| L(s) = 1 | − 4·2-s + 2·3-s + 12·4-s − 10·5-s − 8·6-s − 32·8-s − 6·9-s + 40·10-s + 54·11-s + 24·12-s − 26·13-s − 20·15-s + 80·16-s + 68·17-s + 24·18-s − 42·19-s − 120·20-s − 216·22-s + 294·23-s − 64·24-s + 75·25-s + 104·26-s + 22·27-s + 312·29-s + 80·30-s + 170·31-s − 192·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.384·3-s + 3/2·4-s − 0.894·5-s − 0.544·6-s − 1.41·8-s − 2/9·9-s + 1.26·10-s + 1.48·11-s + 0.577·12-s − 0.554·13-s − 0.344·15-s + 5/4·16-s + 0.970·17-s + 0.314·18-s − 0.507·19-s − 1.34·20-s − 2.09·22-s + 2.66·23-s − 0.544·24-s + 3/5·25-s + 0.784·26-s + 0.156·27-s + 1.99·29-s + 0.486·30-s + 0.984·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.105711654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.105711654\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 366 T^{2} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 54 T + 3266 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 p T + 9702 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 42 T - 3246 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 294 T + 45538 T^{2} - 294 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 312 T + 72134 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 170 T + 66762 T^{2} - 170 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 408 T + 137142 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 140 T + 113862 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 322 T + 183130 T^{2} + 322 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 184 T + 157790 T^{2} + 184 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 116 T + 22638 T^{2} + 116 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 382 T + 304434 T^{2} + 382 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 48 T + 389558 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1620 T + 1220646 T^{2} + 1620 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 930 T + 910922 T^{2} - 930 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 456 T + 609518 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 532 T + 950254 T^{2} - 532 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 120 T - 163546 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 292 T + 1261974 T^{2} - 292 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1092 T + 2030982 T^{2} + 1092 p^{3} T^{3} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.83276631086313960662308754031, −12.33928374797789475297987492169, −11.79282832997231467249648953446, −11.71440749110991913760591008001, −10.78126811038209967677185000581, −10.60199092606174786114806412270, −9.812266539307662815165346691800, −9.354646774790104184859736613312, −8.729931245950586009128807209072, −8.507688877116311488866153316027, −7.911299962931347322462937407290, −7.22976605035331097199902423884, −6.70621648449117289917079969740, −6.40030023212179169499896846661, −5.11925627319002844394965229553, −4.49796171678336223183279309291, −3.20642422421071249631487056394, −3.05089722318831834150428472153, −1.53334450030353088230815352450, −0.72000341580046178825158662972,
0.72000341580046178825158662972, 1.53334450030353088230815352450, 3.05089722318831834150428472153, 3.20642422421071249631487056394, 4.49796171678336223183279309291, 5.11925627319002844394965229553, 6.40030023212179169499896846661, 6.70621648449117289917079969740, 7.22976605035331097199902423884, 7.911299962931347322462937407290, 8.507688877116311488866153316027, 8.729931245950586009128807209072, 9.354646774790104184859736613312, 9.812266539307662815165346691800, 10.60199092606174786114806412270, 10.78126811038209967677185000581, 11.71440749110991913760591008001, 11.79282832997231467249648953446, 12.33928374797789475297987492169, 12.83276631086313960662308754031
