| L(s) = 1 | − 6·2-s − 2·3-s + 21·4-s − 2·5-s + 12·6-s + 3·7-s − 56·8-s − 8·9-s + 12·10-s − 6·11-s − 42·12-s − 9·13-s − 18·14-s + 4·15-s + 126·16-s − 17·17-s + 48·18-s + 19-s − 42·20-s − 6·21-s + 36·22-s − 20·23-s + 112·24-s − 8·25-s + 54·26-s + 23·27-s + 63·28-s + ⋯ |
| L(s) = 1 | − 4.24·2-s − 1.15·3-s + 21/2·4-s − 0.894·5-s + 4.89·6-s + 1.13·7-s − 19.7·8-s − 8/3·9-s + 3.79·10-s − 1.80·11-s − 12.1·12-s − 2.49·13-s − 4.81·14-s + 1.03·15-s + 63/2·16-s − 4.12·17-s + 11.3·18-s + 0.229·19-s − 9.39·20-s − 1.30·21-s + 7.67·22-s − 4.17·23-s + 22.8·24-s − 8/5·25-s + 10.5·26-s + 4.42·27-s + 11.9·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 431^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 431^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T )^{6} \) |
| 431 | \( ( 1 + T )^{6} \) |
| good | 3 | \( 1 + 2 T + 4 p T^{2} + 17 T^{3} + 22 p T^{4} + 73 T^{5} + 233 T^{6} + 73 p T^{7} + 22 p^{3} T^{8} + 17 p^{3} T^{9} + 4 p^{5} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 5 | \( 1 + 2 T + 12 T^{2} + 6 p T^{3} + 24 p T^{4} + 49 p T^{5} + 661 T^{6} + 49 p^{2} T^{7} + 24 p^{3} T^{8} + 6 p^{4} T^{9} + 12 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 7 | \( 1 - 3 T + 29 T^{2} - p^{2} T^{3} + 310 T^{4} - 39 p T^{5} + 2182 T^{6} - 39 p^{2} T^{7} + 310 p^{2} T^{8} - p^{5} T^{9} + 29 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 + 6 T + 48 T^{2} + 227 T^{3} + 1124 T^{4} + 4111 T^{5} + 15599 T^{6} + 4111 p T^{7} + 1124 p^{2} T^{8} + 227 p^{3} T^{9} + 48 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 + 9 T + 69 T^{2} + 347 T^{3} + 1604 T^{4} + 6061 T^{5} + 22910 T^{6} + 6061 p T^{7} + 1604 p^{2} T^{8} + 347 p^{3} T^{9} + 69 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 + p T + 196 T^{2} + 1612 T^{3} + 10684 T^{4} + 57492 T^{5} + 259744 T^{6} + 57492 p T^{7} + 10684 p^{2} T^{8} + 1612 p^{3} T^{9} + 196 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 - T + 72 T^{2} - 92 T^{3} + 2710 T^{4} - 3364 T^{5} + 63083 T^{6} - 3364 p T^{7} + 2710 p^{2} T^{8} - 92 p^{3} T^{9} + 72 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 + 20 T + 244 T^{2} + 2156 T^{3} + 15310 T^{4} + 91015 T^{5} + 468821 T^{6} + 91015 p T^{7} + 15310 p^{2} T^{8} + 2156 p^{3} T^{9} + 244 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 - T + 161 T^{2} - 121 T^{3} + 11109 T^{4} - 6338 T^{5} + 422299 T^{6} - 6338 p T^{7} + 11109 p^{2} T^{8} - 121 p^{3} T^{9} + 161 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 - 8 T + 137 T^{2} - 861 T^{3} + 8592 T^{4} - 43792 T^{5} + 330326 T^{6} - 43792 p T^{7} + 8592 p^{2} T^{8} - 861 p^{3} T^{9} + 137 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( 1 + 3 T + 142 T^{2} + 374 T^{3} + 10298 T^{4} + 22186 T^{5} + 466616 T^{6} + 22186 p T^{7} + 10298 p^{2} T^{8} + 374 p^{3} T^{9} + 142 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( 1 + 19 T + 330 T^{2} + 3913 T^{3} + 38572 T^{4} + 321343 T^{5} + 2191812 T^{6} + 321343 p T^{7} + 38572 p^{2} T^{8} + 3913 p^{3} T^{9} + 330 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 21 T + 368 T^{2} + 4390 T^{3} + 45032 T^{4} + 371896 T^{5} + 2674920 T^{6} + 371896 p T^{7} + 45032 p^{2} T^{8} + 4390 p^{3} T^{9} + 368 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 + 10 T + 159 T^{2} + 459 T^{3} + 2364 T^{4} - 54006 T^{5} - 266724 T^{6} - 54006 p T^{7} + 2364 p^{2} T^{8} + 459 p^{3} T^{9} + 159 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 + T + 62 T^{2} + 602 T^{3} + 3056 T^{4} + 7280 T^{5} + 368223 T^{6} + 7280 p T^{7} + 3056 p^{2} T^{8} + 602 p^{3} T^{9} + 62 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 - 11 T + 197 T^{2} - 1413 T^{3} + 16943 T^{4} - 104080 T^{5} + 1099549 T^{6} - 104080 p T^{7} + 16943 p^{2} T^{8} - 1413 p^{3} T^{9} + 197 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 + 2 T + 223 T^{2} - 375 T^{3} + 18660 T^{4} - 107712 T^{5} + 1070434 T^{6} - 107712 p T^{7} + 18660 p^{2} T^{8} - 375 p^{3} T^{9} + 223 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 4 T + 271 T^{2} - 541 T^{3} + 34378 T^{4} - 41338 T^{5} + 2812384 T^{6} - 41338 p T^{7} + 34378 p^{2} T^{8} - 541 p^{3} T^{9} + 271 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 + 4 T + 291 T^{2} + 344 T^{3} + 35121 T^{4} - 29127 T^{5} + 2768778 T^{6} - 29127 p T^{7} + 35121 p^{2} T^{8} + 344 p^{3} T^{9} + 291 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 + 39 T + 800 T^{2} + 12316 T^{3} + 158260 T^{4} + 1697904 T^{5} + 15536962 T^{6} + 1697904 p T^{7} + 158260 p^{2} T^{8} + 12316 p^{3} T^{9} + 800 p^{4} T^{10} + 39 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 + 16 T + 497 T^{2} + 5533 T^{3} + 96150 T^{4} + 803604 T^{5} + 9972182 T^{6} + 803604 p T^{7} + 96150 p^{2} T^{8} + 5533 p^{3} T^{9} + 497 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 + 20 T + 464 T^{2} + 5493 T^{3} + 76301 T^{4} + 673726 T^{5} + 7474786 T^{6} + 673726 p T^{7} + 76301 p^{2} T^{8} + 5493 p^{3} T^{9} + 464 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 + 17 T + 441 T^{2} + 5158 T^{3} + 75003 T^{4} + 691470 T^{5} + 7774914 T^{6} + 691470 p T^{7} + 75003 p^{2} T^{8} + 5158 p^{3} T^{9} + 441 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 + 29 T + 719 T^{2} + 12621 T^{3} + 187943 T^{4} + 2306480 T^{5} + 24765083 T^{6} + 2306480 p T^{7} + 187943 p^{2} T^{8} + 12621 p^{3} T^{9} + 719 p^{4} T^{10} + 29 p^{5} T^{11} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.16498129826795321633908221129, −5.68905175697511286670677648603, −5.58078163031941193791032685974, −5.42777792614743047301960658772, −5.38280750115608869363478336265, −5.33694842363687822896731267531, −5.14906760713632270232202531891, −4.76405810025956420729923963557, −4.65568393819433903031570293196, −4.32939321457502375273215653606, −4.28488126722566791869080286625, −4.07452640546925442579104746794, −4.04794207511389660312637262566, −3.44081404352532941916372308798, −3.04020241846684506857939261092, −2.95675840162826184742609675172, −2.87004990380342159909739875704, −2.85688414970249145040367873531, −2.72382706081070357360295950535, −2.18654100231582988810685617707, −2.08502863555490571779601371318, −1.84495215660545507175918725981, −1.80413799333512140313693570707, −1.69916984989903885174403769138, −1.27770090329830642501603474967, 0, 0, 0, 0, 0, 0,
1.27770090329830642501603474967, 1.69916984989903885174403769138, 1.80413799333512140313693570707, 1.84495215660545507175918725981, 2.08502863555490571779601371318, 2.18654100231582988810685617707, 2.72382706081070357360295950535, 2.85688414970249145040367873531, 2.87004990380342159909739875704, 2.95675840162826184742609675172, 3.04020241846684506857939261092, 3.44081404352532941916372308798, 4.04794207511389660312637262566, 4.07452640546925442579104746794, 4.28488126722566791869080286625, 4.32939321457502375273215653606, 4.65568393819433903031570293196, 4.76405810025956420729923963557, 5.14906760713632270232202531891, 5.33694842363687822896731267531, 5.38280750115608869363478336265, 5.42777792614743047301960658772, 5.58078163031941193791032685974, 5.68905175697511286670677648603, 6.16498129826795321633908221129
Plot not available for L-functions of degree greater than 10.
