| L(s) = 1 | − 32·2-s − 243·3-s + 1.02e3·4-s − 3.12e3·5-s + 7.77e3·6-s − 2.28e4·7-s − 3.27e4·8-s + 5.90e4·9-s + 1.00e5·10-s − 2.59e5·11-s − 2.48e5·12-s − 2.32e6·13-s + 7.32e5·14-s + 7.59e5·15-s + 1.04e6·16-s − 1.91e6·17-s − 1.88e6·18-s − 1.61e6·19-s − 3.20e6·20-s + 5.55e6·21-s + 8.29e6·22-s + 3.28e7·23-s + 7.96e6·24-s + 9.76e6·25-s + 7.43e7·26-s − 1.43e7·27-s − 2.34e7·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.514·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.485·11-s − 0.288·12-s − 1.73·13-s + 0.363·14-s + 0.258·15-s + 1/4·16-s − 0.327·17-s − 0.235·18-s − 0.149·19-s − 0.223·20-s + 0.297·21-s + 0.343·22-s + 1.06·23-s + 0.204·24-s + 1/5·25-s + 1.22·26-s − 0.192·27-s − 0.257·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.6147915608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6147915608\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{5} T \) |
| 3 | \( 1 + p^{5} T \) |
| 5 | \( 1 + p^{5} T \) |
| good | 7 | \( 1 + 3268 p T + p^{11} T^{2} \) |
| 11 | \( 1 + 259128 T + p^{11} T^{2} \) |
| 13 | \( 1 + 2323462 T + p^{11} T^{2} \) |
| 17 | \( 1 + 1918266 T + p^{11} T^{2} \) |
| 19 | \( 1 + 1613740 T + p^{11} T^{2} \) |
| 23 | \( 1 - 32849208 T + p^{11} T^{2} \) |
| 29 | \( 1 - 152590290 T + p^{11} T^{2} \) |
| 31 | \( 1 - 56810072 T + p^{11} T^{2} \) |
| 37 | \( 1 - 571517354 T + p^{11} T^{2} \) |
| 41 | \( 1 - 734063082 T + p^{11} T^{2} \) |
| 43 | \( 1 + 580696612 T + p^{11} T^{2} \) |
| 47 | \( 1 - 478846584 T + p^{11} T^{2} \) |
| 53 | \( 1 + 2553787902 T + p^{11} T^{2} \) |
| 59 | \( 1 - 6317920440 T + p^{11} T^{2} \) |
| 61 | \( 1 + 786138178 T + p^{11} T^{2} \) |
| 67 | \( 1 + 21287683396 T + p^{11} T^{2} \) |
| 71 | \( 1 - 11719480032 T + p^{11} T^{2} \) |
| 73 | \( 1 - 23175633458 T + p^{11} T^{2} \) |
| 79 | \( 1 - 2698969640 T + p^{11} T^{2} \) |
| 83 | \( 1 - 32658710748 T + p^{11} T^{2} \) |
| 89 | \( 1 - 74556330210 T + p^{11} T^{2} \) |
| 97 | \( 1 - 41434868834 T + p^{11} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.81286159927826928936801083914, −12.88332622629114958687141698499, −11.84637099190547518994686994506, −10.56379617623833385569862968324, −9.449601670584493376831451601766, −7.78671241662167182928633981063, −6.60870957944901872558320923620, −4.82762000785244880135770369650, −2.67612481489988158135922709810, −0.57430365646275643503559246430,
0.57430365646275643503559246430, 2.67612481489988158135922709810, 4.82762000785244880135770369650, 6.60870957944901872558320923620, 7.78671241662167182928633981063, 9.449601670584493376831451601766, 10.56379617623833385569862968324, 11.84637099190547518994686994506, 12.88332622629114958687141698499, 14.81286159927826928936801083914
