| L(s) = 1 | + 7.28·2-s − 14.9·3-s + 21.0·4-s − 109.·6-s + 135.·7-s − 79.9·8-s − 18.1·9-s + 191.·11-s − 315.·12-s + 169·13-s + 985.·14-s − 1.25e3·16-s − 874.·17-s − 132.·18-s + 1.99e3·19-s − 2.02e3·21-s + 1.39e3·22-s − 2.09e3·23-s + 1.19e3·24-s + 1.23e3·26-s + 3.91e3·27-s + 2.84e3·28-s − 46.3·29-s − 9.45e3·31-s − 6.57e3·32-s − 2.86e3·33-s − 6.36e3·34-s + ⋯ |
| L(s) = 1 | + 1.28·2-s − 0.961·3-s + 0.656·4-s − 1.23·6-s + 1.04·7-s − 0.441·8-s − 0.0748·9-s + 0.476·11-s − 0.631·12-s + 0.277·13-s + 1.34·14-s − 1.22·16-s − 0.733·17-s − 0.0963·18-s + 1.26·19-s − 1.00·21-s + 0.613·22-s − 0.824·23-s + 0.424·24-s + 0.357·26-s + 1.03·27-s + 0.686·28-s − 0.0102·29-s − 1.76·31-s − 1.13·32-s − 0.458·33-s − 0.944·34-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 | \( 1 - 7.28T + 32T^{2} \) |
| 3 | \( 1 + 14.9T + 243T^{2} \) |
| 7 | \( 1 - 135.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 191.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 874.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.99e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.09e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 46.3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.45e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.37e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.05e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.05e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.70e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.39e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.01e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.74e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 317.T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.96e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.76e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.45e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.42e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.32e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.93e4T + 8.58e9T^{2} \) |
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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
−10.90441037287139051380037697439, −9.424813627840420509932129328744, −8.336812510566197443666915274850, −6.99380442961686993821610266195, −5.97277577034537295249617878889, −5.25532999502275137010292402118, −4.48121316880551093515981470492, −3.30989520344360956787539584261, −1.67704801071822074078879513202, 0,
1.67704801071822074078879513202, 3.30989520344360956787539584261, 4.48121316880551093515981470492, 5.25532999502275137010292402118, 5.97277577034537295249617878889, 6.99380442961686993821610266195, 8.336812510566197443666915274850, 9.424813627840420509932129328744, 10.90441037287139051380037697439
