| L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s − 84.5·5-s + 36·6-s + 83.0·7-s + 64·8-s + 81·9-s − 338.·10-s + 426.·11-s + 144·12-s − 689.·13-s + 332.·14-s − 761.·15-s + 256·16-s − 1.68e3·17-s + 324·18-s − 1.96e3·19-s − 1.35e3·20-s + 747.·21-s + 1.70e3·22-s + 330.·23-s + 576·24-s + 4.02e3·25-s − 2.75e3·26-s + 729·27-s + 1.32e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.51·5-s + 0.408·6-s + 0.640·7-s + 0.353·8-s + 0.333·9-s − 1.06·10-s + 1.06·11-s + 0.288·12-s − 1.13·13-s + 0.453·14-s − 0.873·15-s + 0.250·16-s − 1.41·17-s + 0.235·18-s − 1.24·19-s − 0.756·20-s + 0.369·21-s + 0.752·22-s + 0.130·23-s + 0.204·24-s + 1.28·25-s − 0.800·26-s + 0.192·27-s + 0.320·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - 4T \) |
| 3 | \( 1 - 9T \) |
| 59 | \( 1 + 3.48e3T \) |
| good | 5 | \( 1 + 84.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 83.0T + 1.68e4T^{2} \) |
| 11 | \( 1 - 426.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 689.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.68e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.96e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 330.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 550.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.53e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.36e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.31e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.04e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 5.06e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.94e3T + 4.18e8T^{2} \) |
| 61 | \( 1 + 3.92e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 7.26e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.11e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.38e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.34e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.49e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.44e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.67e4T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.42872577668026266612323455626, −8.969385306622673637527039903852, −8.237342205299011917852791368466, −7.28802151114230545849724258566, −6.52028153036139566192094913163, −4.60841494065760044968979129227, −4.33497692010369620822392147466, −3.11145027741474979926795041331, −1.80518080389181619280320621378, 0,
1.80518080389181619280320621378, 3.11145027741474979926795041331, 4.33497692010369620822392147466, 4.60841494065760044968979129227, 6.52028153036139566192094913163, 7.28802151114230545849724258566, 8.237342205299011917852791368466, 8.969385306622673637527039903852, 10.42872577668026266612323455626
