| L(s) = 1 | + 0.322·2-s − 9·3-s − 31.8·4-s − 76.7·5-s − 2.90·6-s + 42.9·7-s − 20.6·8-s + 81·9-s − 24.7·10-s − 138.·11-s + 287.·12-s + 1.07e3·13-s + 13.8·14-s + 690.·15-s + 1.01e3·16-s − 1.96e3·17-s + 26.1·18-s + 1.51e3·19-s + 2.44e3·20-s − 386.·21-s − 44.8·22-s + 3.01e3·23-s + 185.·24-s + 2.76e3·25-s + 345.·26-s − 729·27-s − 1.37e3·28-s + ⋯ |
| L(s) = 1 | + 0.0570·2-s − 0.577·3-s − 0.996·4-s − 1.37·5-s − 0.0329·6-s + 0.331·7-s − 0.113·8-s + 0.333·9-s − 0.0783·10-s − 0.345·11-s + 0.575·12-s + 1.75·13-s + 0.0189·14-s + 0.792·15-s + 0.990·16-s − 1.65·17-s + 0.0190·18-s + 0.965·19-s + 1.36·20-s − 0.191·21-s − 0.0197·22-s + 1.18·23-s + 0.0657·24-s + 0.884·25-s + 0.100·26-s − 0.192·27-s − 0.330·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 9T \) |
| 103 | \( 1 + 1.06e4T \) |
| good | 2 | \( 1 - 0.322T + 32T^{2} \) |
| 5 | \( 1 + 76.7T + 3.12e3T^{2} \) |
| 7 | \( 1 - 42.9T + 1.68e4T^{2} \) |
| 11 | \( 1 + 138.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.07e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.96e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.51e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.01e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.18e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.81e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.26e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.17e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.63e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.16e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.81e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.14e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.29e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.65e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.15e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.22e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.65e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.49e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.06e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.50e5T + 8.58e9T^{2} \) |
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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
−10.74171157563122342851684613006, −9.244392220905108194278138505001, −8.471643727251437049523301041265, −7.65032447909516254895848702624, −6.41020785085652116524597903448, −5.11500212985369135959431814499, −4.28635040467634930770603990652, −3.39452142818459041277148790975, −1.08375905582519049150119093846, 0,
1.08375905582519049150119093846, 3.39452142818459041277148790975, 4.28635040467634930770603990652, 5.11500212985369135959431814499, 6.41020785085652116524597903448, 7.65032447909516254895848702624, 8.471643727251437049523301041265, 9.244392220905108194278138505001, 10.74171157563122342851684613006
