| L(s) = 1 | + 4·2-s + 6·3-s + 12·4-s − 20·5-s + 24·6-s − 23·7-s + 32·8-s + 27·9-s − 80·10-s − 80·11-s + 72·12-s − 102·13-s − 92·14-s − 120·15-s + 80·16-s − 25·17-s + 108·18-s − 55·19-s − 240·20-s − 138·21-s − 320·22-s − 5·23-s + 192·24-s + 63·25-s − 408·26-s + 108·27-s − 276·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.78·5-s + 1.63·6-s − 1.24·7-s + 1.41·8-s + 9-s − 2.52·10-s − 2.19·11-s + 1.73·12-s − 2.17·13-s − 1.75·14-s − 2.06·15-s + 5/4·16-s − 0.356·17-s + 1.41·18-s − 0.664·19-s − 2.68·20-s − 1.43·21-s − 3.10·22-s − 0.0453·23-s + 1.63·24-s + 0.503·25-s − 3.07·26-s + 0.769·27-s − 1.86·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125316 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 59 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 4 p T + 337 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 23 T + 555 T^{2} + 23 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 80 T + 3937 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 102 T + 5695 T^{2} + 102 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 25 T + 9953 T^{2} + 25 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 55 T + 14471 T^{2} + 55 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5 T - 6239 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 35 T + 45545 T^{2} + 35 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 53 T - 8047 T^{2} + 53 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 221 T + 105713 T^{2} + 221 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 89 T + 5893 T^{2} + 89 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 508 T + 194813 T^{2} + 508 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 579 T + 202975 T^{2} - 579 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 432 T + 189957 T^{2} - 432 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 151 T + 439379 T^{2} - 151 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 168 T + 192049 T^{2} + 168 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 532 T + 692653 T^{2} - 532 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 49 T + 731051 T^{2} + 49 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 664 T + 1030769 T^{2} + 664 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 351 T + 667293 T^{2} + 351 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1401 T + 1826535 T^{2} + 1401 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 452 T + 1194689 T^{2} - 452 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.80835653439212612808062116554, −10.40726245870704875559595708517, −9.870233400072842743784402405470, −9.745140274102151914456190099635, −8.617453306818629703377088569207, −8.428202999746882421239929726283, −7.61883911368111790076843351312, −7.57626833930540170745400011956, −7.08594403243481762508455341600, −6.70166969268097947378253937646, −5.69185106854829735864340003064, −5.22439331483275777344699861789, −4.59020574929205439176871432434, −4.22950515213359610547220923063, −3.40189163156350408354148911438, −3.28723322988448770571983106083, −2.33173747463694666456213607619, −2.32853261818182609881162613003, 0, 0,
2.32853261818182609881162613003, 2.33173747463694666456213607619, 3.28723322988448770571983106083, 3.40189163156350408354148911438, 4.22950515213359610547220923063, 4.59020574929205439176871432434, 5.22439331483275777344699861789, 5.69185106854829735864340003064, 6.70166969268097947378253937646, 7.08594403243481762508455341600, 7.57626833930540170745400011956, 7.61883911368111790076843351312, 8.428202999746882421239929726283, 8.617453306818629703377088569207, 9.745140274102151914456190099635, 9.870233400072842743784402405470, 10.40726245870704875559595708517, 10.80835653439212612808062116554
