| L(s) = 1 | − 128·2-s + 6.41e3·3-s + 1.63e4·4-s − 2.78e5·5-s − 8.21e5·6-s − 2.09e6·8-s + 2.68e7·9-s + 3.57e7·10-s + 1.07e8·11-s + 1.05e8·12-s − 3.29e8·13-s − 1.79e9·15-s + 2.68e8·16-s − 4.56e8·17-s − 3.43e9·18-s + 3.54e9·19-s − 4.57e9·20-s − 1.37e10·22-s − 3.01e10·23-s − 1.34e10·24-s + 4.73e10·25-s + 4.21e10·26-s + 8.00e10·27-s + 3.19e10·29-s + 2.29e11·30-s + 1.76e11·31-s − 3.43e10·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.69·3-s + 0.5·4-s − 1.59·5-s − 1.19·6-s − 0.353·8-s + 1.86·9-s + 1.12·10-s + 1.65·11-s + 0.847·12-s − 1.45·13-s − 2.70·15-s + 0.250·16-s − 0.269·17-s − 1.32·18-s + 0.909·19-s − 0.798·20-s − 1.17·22-s − 1.84·23-s − 0.598·24-s + 1.55·25-s + 1.03·26-s + 1.47·27-s + 0.343·29-s + 1.91·30-s + 1.15·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 128T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 6.41e3T + 1.43e7T^{2} \) |
| 5 | \( 1 + 2.78e5T + 3.05e10T^{2} \) |
| 11 | \( 1 - 1.07e8T + 4.17e15T^{2} \) |
| 13 | \( 1 + 3.29e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 4.56e8T + 2.86e18T^{2} \) |
| 19 | \( 1 - 3.54e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + 3.01e10T + 2.66e20T^{2} \) |
| 29 | \( 1 - 3.19e10T + 8.62e21T^{2} \) |
| 31 | \( 1 - 1.76e11T + 2.34e22T^{2} \) |
| 37 | \( 1 + 4.15e11T + 3.33e23T^{2} \) |
| 41 | \( 1 - 4.32e11T + 1.55e24T^{2} \) |
| 43 | \( 1 + 2.09e12T + 3.17e24T^{2} \) |
| 47 | \( 1 - 3.73e12T + 1.20e25T^{2} \) |
| 53 | \( 1 + 8.00e10T + 7.31e25T^{2} \) |
| 59 | \( 1 + 9.04e12T + 3.65e26T^{2} \) |
| 61 | \( 1 - 8.15e11T + 6.02e26T^{2} \) |
| 67 | \( 1 + 7.11e12T + 2.46e27T^{2} \) |
| 71 | \( 1 + 5.65e12T + 5.87e27T^{2} \) |
| 73 | \( 1 + 6.12e12T + 8.90e27T^{2} \) |
| 79 | \( 1 + 2.35e14T + 2.91e28T^{2} \) |
| 83 | \( 1 - 9.34e13T + 6.11e28T^{2} \) |
| 89 | \( 1 + 2.13e14T + 1.74e29T^{2} \) |
| 97 | \( 1 - 6.81e14T + 6.33e29T^{2} \) |
| 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
−10.06743466772248478942111304600, −9.227756808582221459388979890343, −8.311257595577329948426541155119, −7.63880411465809946841325170954, −6.83193008353447144439638970391, −4.35836298366798242971174314577, −3.61068333839163777226679627468, −2.57299428018996845294255951132, −1.34350189096015369917632530818, 0,
1.34350189096015369917632530818, 2.57299428018996845294255951132, 3.61068333839163777226679627468, 4.35836298366798242971174314577, 6.83193008353447144439638970391, 7.63880411465809946841325170954, 8.311257595577329948426541155119, 9.227756808582221459388979890343, 10.06743466772248478942111304600
