| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s − 8.96·5-s − 36·6-s − 187.·7-s − 64·8-s + 81·9-s + 35.8·10-s − 189.·11-s + 144·12-s + 737.·13-s + 751.·14-s − 80.6·15-s + 256·16-s + 388.·17-s − 324·18-s + 1.78e3·19-s − 143.·20-s − 1.69e3·21-s + 756.·22-s + 4.32e3·23-s − 576·24-s − 3.04e3·25-s − 2.94e3·26-s + 729·27-s − 3.00e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.160·5-s − 0.408·6-s − 1.44·7-s − 0.353·8-s + 0.333·9-s + 0.113·10-s − 0.471·11-s + 0.288·12-s + 1.20·13-s + 1.02·14-s − 0.0925·15-s + 0.250·16-s + 0.326·17-s − 0.235·18-s + 1.13·19-s − 0.0801·20-s − 0.836·21-s + 0.333·22-s + 1.70·23-s − 0.204·24-s − 0.974·25-s − 0.855·26-s + 0.192·27-s − 0.724·28-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 | \( 1 + 8.96T + 3.12e3T^{2} \) |
| 7 | \( 1 + 187.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 189.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 737.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 388.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.78e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.32e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.03e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.53e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.11e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.76e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.74e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 4.36e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.26e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 4.88e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.18e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 119.T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.36e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.40e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.62e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.07e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.67e4T + 8.58e9T^{2} \) |
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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
−9.919618902059782363593320441030, −9.312632759412355256081583817375, −8.439955037674955774993626406120, −7.41502270430330468859676541300, −6.59026895707659445513488400397, −5.45275395444713408873862838896, −3.58277703078417264218515264763, −3.00042488676917252462923989471, −1.37961228764649211084540485846, 0,
1.37961228764649211084540485846, 3.00042488676917252462923989471, 3.58277703078417264218515264763, 5.45275395444713408873862838896, 6.59026895707659445513488400397, 7.41502270430330468859676541300, 8.439955037674955774993626406120, 9.312632759412355256081583817375, 9.919618902059782363593320441030
