| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s + 25·5-s − 36·6-s − 34.7·7-s − 64·8-s + 81·9-s − 100·10-s − 728.·11-s + 144·12-s − 66.1·13-s + 139.·14-s + 225·15-s + 256·16-s + 2.03e3·17-s − 324·18-s − 361·19-s + 400·20-s − 313.·21-s + 2.91e3·22-s + 3.34e3·23-s − 576·24-s + 625·25-s + 264.·26-s + 729·27-s − 556.·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 0.268·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 1.81·11-s + 0.288·12-s − 0.108·13-s + 0.189·14-s + 0.258·15-s + 0.250·16-s + 1.70·17-s − 0.235·18-s − 0.229·19-s + 0.223·20-s − 0.154·21-s + 1.28·22-s + 1.32·23-s − 0.204·24-s + 0.200·25-s + 0.0767·26-s + 0.192·27-s − 0.134·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 9T \) |
| 5 | \( 1 - 25T \) |
| 19 | \( 1 + 361T \) |
| good | 7 | \( 1 + 34.7T + 1.68e4T^{2} \) |
| 11 | \( 1 + 728.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 66.1T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.03e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 3.34e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.95e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.30e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.99e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.33e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.59e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.53e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.11e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 7.04e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.60e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.84e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.22e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.20e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.37e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.39e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.21e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.74e4T + 8.58e9T^{2} \) |
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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
−9.580313849419220827499080934155, −8.647280926512718012623473102719, −7.77541710092852140771390855033, −7.19336307486245037741133375246, −5.84496618749708180461558501601, −5.03644006949666833169510580551, −3.31940564796840064290209000180, −2.59148080287552847768976901930, −1.37653778158877842622807619696, 0,
1.37653778158877842622807619696, 2.59148080287552847768976901930, 3.31940564796840064290209000180, 5.03644006949666833169510580551, 5.84496618749708180461558501601, 7.19336307486245037741133375246, 7.77541710092852140771390855033, 8.647280926512718012623473102719, 9.580313849419220827499080934155
