| L(s) = 1 | + 4.30·2-s + 19.9·3-s − 13.4·4-s + 85.7·6-s + 125.·7-s − 195.·8-s + 153.·9-s − 709.·11-s − 267.·12-s + 169·13-s + 541.·14-s − 412.·16-s − 1.92e3·17-s + 660.·18-s − 2.16e3·19-s + 2.50e3·21-s − 3.05e3·22-s − 305.·23-s − 3.89e3·24-s + 727.·26-s − 1.78e3·27-s − 1.69e3·28-s + 4.51e3·29-s + 4.07e3·31-s + 4.48e3·32-s − 1.41e4·33-s − 8.29e3·34-s + ⋯ |
| L(s) = 1 | + 0.761·2-s + 1.27·3-s − 0.420·4-s + 0.972·6-s + 0.970·7-s − 1.08·8-s + 0.631·9-s − 1.76·11-s − 0.537·12-s + 0.277·13-s + 0.738·14-s − 0.402·16-s − 1.61·17-s + 0.480·18-s − 1.37·19-s + 1.23·21-s − 1.34·22-s − 0.120·23-s − 1.38·24-s + 0.211·26-s − 0.471·27-s − 0.408·28-s + 0.996·29-s + 0.762·31-s + 0.774·32-s − 2.25·33-s − 1.23·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 - 169T \) |
| good | 2 | \( 1 - 4.30T + 32T^{2} \) |
| 3 | \( 1 - 19.9T + 243T^{2} \) |
| 7 | \( 1 - 125.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 709.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 1.92e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.16e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 305.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.51e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.07e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.23e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.44e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.71e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.83e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.19e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.98e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.49e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.04e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.45e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.90e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.33e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.38e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.24e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.38e4T + 8.58e9T^{2} \) |
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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
−10.35699232180373174201953123910, −9.009129930207136535761959963943, −8.453739483802988699528983485175, −7.77605092608974429902220953248, −6.23015161783151238353249644833, −4.91611368465024590704238974851, −4.28006057080907792511649350576, −2.92274467689949341862851432910, −2.13332138496309952487703253777, 0,
2.13332138496309952487703253777, 2.92274467689949341862851432910, 4.28006057080907792511649350576, 4.91611368465024590704238974851, 6.23015161783151238353249644833, 7.77605092608974429902220953248, 8.453739483802988699528983485175, 9.009129930207136535761959963943, 10.35699232180373174201953123910
