| L(s) = 1 | − 10·3-s − 10·5-s + 43·9-s + 52·11-s + 52·13-s + 100·15-s + 76·17-s − 176·19-s + 102·23-s + 75·25-s − 130·27-s − 306·29-s − 244·31-s − 520·33-s − 176·37-s − 520·39-s + 130·41-s + 114·43-s − 430·45-s − 716·47-s − 760·51-s + 808·53-s − 520·55-s + 1.76e3·57-s + 1.01e3·59-s + 222·61-s − 520·65-s + ⋯ |
| L(s) = 1 | − 1.92·3-s − 0.894·5-s + 1.59·9-s + 1.42·11-s + 1.10·13-s + 1.72·15-s + 1.08·17-s − 2.12·19-s + 0.924·23-s + 3/5·25-s − 0.926·27-s − 1.95·29-s − 1.41·31-s − 2.74·33-s − 0.782·37-s − 2.13·39-s + 0.495·41-s + 0.404·43-s − 1.42·45-s − 2.22·47-s − 2.08·51-s + 2.09·53-s − 1.27·55-s + 4.08·57-s + 2.24·59-s + 0.465·61-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.178575606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.178575606\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 10 T + 19 p T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 52 T + 3250 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 p T + 758 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 76 T + 6958 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 176 T + 21374 T^{2} + 176 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 102 T + 26737 T^{2} - 102 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 306 T + 69019 T^{2} + 306 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 244 T + 71298 T^{2} + 244 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 176 T + 58362 T^{2} + 176 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 130 T + 57499 T^{2} - 130 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 114 T - 36287 T^{2} - 114 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 716 T + 327010 T^{2} + 716 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 808 T + 450322 T^{2} - 808 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1016 T + 637054 T^{2} - 1016 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 222 T + 271891 T^{2} - 222 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 p T + 544217 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 296 T + 268774 T^{2} - 296 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 724 T + 414078 T^{2} + 724 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1128 T + 1156246 T^{2} - 1128 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 338 T + 402817 T^{2} + 338 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 326 T + 371707 T^{2} - 326 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 20 p T + 2737734 T^{2} - 20 p^{4} T^{3} + p^{6} T^{4} \) |
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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
−9.954755348543484654836900129820, −9.409373243879251604409366051527, −8.908616059225597511517855512717, −8.674076799664625407160598278834, −8.204116343659573844174204884722, −7.59067111041127478372050368899, −6.99383898192049310173795933768, −6.95275028514484982735669260386, −6.16846326104210229749981260855, −6.14669677238116089122543209814, −5.42844977848105112715197506725, −5.31200601935417763893548870898, −4.54584263055735643542499671926, −4.05939198796782390747310819105, −3.58027738777870968371698908712, −3.46919837289302808021341901190, −2.07096186354082263208705660101, −1.64997234998782516868064760797, −0.67916213214335271252048488797, −0.52153527677845384586573473499,
0.52153527677845384586573473499, 0.67916213214335271252048488797, 1.64997234998782516868064760797, 2.07096186354082263208705660101, 3.46919837289302808021341901190, 3.58027738777870968371698908712, 4.05939198796782390747310819105, 4.54584263055735643542499671926, 5.31200601935417763893548870898, 5.42844977848105112715197506725, 6.14669677238116089122543209814, 6.16846326104210229749981260855, 6.95275028514484982735669260386, 6.99383898192049310173795933768, 7.59067111041127478372050368899, 8.204116343659573844174204884722, 8.674076799664625407160598278834, 8.908616059225597511517855512717, 9.409373243879251604409366051527, 9.954755348543484654836900129820
