| L(s) = 1 | − 3.48·2-s − 0.931·3-s − 19.8·4-s − 25·5-s + 3.24·6-s + 180.·8-s − 242.·9-s + 87.0·10-s + 286.·11-s + 18.5·12-s + 129.·13-s + 23.2·15-s + 6.61·16-s + 445.·17-s + 843.·18-s + 828.·19-s + 496.·20-s − 998.·22-s − 550.·23-s − 168.·24-s + 625·25-s − 452.·26-s + 451.·27-s + 4.84e3·29-s − 81.0·30-s + 5.80e3·31-s − 5.80e3·32-s + ⋯ |
| L(s) = 1 | − 0.615·2-s − 0.0597·3-s − 0.620·4-s − 0.447·5-s + 0.0367·6-s + 0.997·8-s − 0.996·9-s + 0.275·10-s + 0.714·11-s + 0.0370·12-s + 0.213·13-s + 0.0267·15-s + 0.00645·16-s + 0.373·17-s + 0.613·18-s + 0.526·19-s + 0.277·20-s − 0.439·22-s − 0.216·23-s − 0.0596·24-s + 0.200·25-s − 0.131·26-s + 0.119·27-s + 1.06·29-s − 0.0164·30-s + 1.08·31-s − 1.00·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 25T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3.48T + 32T^{2} \) |
| 3 | \( 1 + 0.931T + 243T^{2} \) |
| 11 | \( 1 - 286.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 129.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 445.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 828.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 550.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.84e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.80e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.40e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.18e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.58e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.03e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.47e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.25e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.19e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.26e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.58e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.00e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.48e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.15e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.20e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.59e5T + 8.58e9T^{2} \) |
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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
−10.63536449867408341366473114286, −9.679507180541610888160367424349, −8.685523828771044272052092357325, −8.136777150917029717575689308008, −6.89064120182838122905440724554, −5.55938079860846576032661275872, −4.38945859348580681424039218890, −3.16600307676482230185090638794, −1.23875696718686591450149867692, 0,
1.23875696718686591450149867692, 3.16600307676482230185090638794, 4.38945859348580681424039218890, 5.55938079860846576032661275872, 6.89064120182838122905440724554, 8.136777150917029717575689308008, 8.685523828771044272052092357325, 9.679507180541610888160367424349, 10.63536449867408341366473114286
