| L(s) = 1 | + 4·2-s − 15.0·3-s + 16·4-s − 60.1·6-s − 52.9·7-s + 64·8-s − 16.7·9-s − 259.·11-s − 240.·12-s + 169·13-s − 211.·14-s + 256·16-s + 2.27e3·17-s − 67.0·18-s + 730.·19-s + 796.·21-s − 1.03e3·22-s − 1.97e3·23-s − 962.·24-s + 676·26-s + 3.90e3·27-s − 847.·28-s − 949.·29-s + 225.·31-s + 1.02e3·32-s + 3.89e3·33-s + 9.09e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.964·3-s + 0.5·4-s − 0.682·6-s − 0.408·7-s + 0.353·8-s − 0.0689·9-s − 0.646·11-s − 0.482·12-s + 0.277·13-s − 0.288·14-s + 0.250·16-s + 1.90·17-s − 0.0487·18-s + 0.464·19-s + 0.394·21-s − 0.456·22-s − 0.777·23-s − 0.341·24-s + 0.196·26-s + 1.03·27-s − 0.204·28-s − 0.209·29-s + 0.0421·31-s + 0.176·32-s + 0.623·33-s + 1.34·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 4T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - 169T \) |
| good | 3 | \( 1 + 15.0T + 243T^{2} \) |
| 7 | \( 1 + 52.9T + 1.68e4T^{2} \) |
| 11 | \( 1 + 259.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 2.27e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 730.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.97e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 949.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 225.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 954.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.73e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.00e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.06e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.61e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 326.T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.68e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.35e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.17e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.32e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.62e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.01e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.67e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.51e5T + 8.58e9T^{2} \) |
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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
−9.661424740767106877540296716837, −8.257915936484532617406275530042, −7.41391273703740005334771461912, −6.32309051944198239145421325964, −5.63810298597729792091611634647, −5.03642640504899894289498140108, −3.70950936553542139256605393018, −2.80484413018033340477116498766, −1.25346453166362685586256091356, 0,
1.25346453166362685586256091356, 2.80484413018033340477116498766, 3.70950936553542139256605393018, 5.03642640504899894289498140108, 5.63810298597729792091611634647, 6.32309051944198239145421325964, 7.41391273703740005334771461912, 8.257915936484532617406275530042, 9.661424740767106877540296716837
