| L(s) = 1 | − 8·2-s + 58.8·3-s + 64·4-s + 424.·5-s − 470.·6-s − 512·8-s + 1.27e3·9-s − 3.39e3·10-s + 7.88e3·11-s + 3.76e3·12-s + 6.71e3·13-s + 2.49e4·15-s + 4.09e3·16-s − 7.07e3·17-s − 1.01e4·18-s − 2.62e4·19-s + 2.71e4·20-s − 6.31e4·22-s + 1.20e4·23-s − 3.01e4·24-s + 1.01e5·25-s − 5.37e4·26-s − 5.38e4·27-s + 3.06e3·29-s − 1.99e5·30-s − 1.18e5·31-s − 3.27e4·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.25·3-s + 0.5·4-s + 1.51·5-s − 0.889·6-s − 0.353·8-s + 0.581·9-s − 1.07·10-s + 1.78·11-s + 0.628·12-s + 0.848·13-s + 1.90·15-s + 0.250·16-s − 0.349·17-s − 0.410·18-s − 0.877·19-s + 0.758·20-s − 1.26·22-s + 0.206·23-s − 0.444·24-s + 1.30·25-s − 0.599·26-s − 0.526·27-s + 0.0233·29-s − 1.34·30-s − 0.714·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.220530128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.220530128\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 58.8T + 2.18e3T^{2} \) |
| 5 | \( 1 - 424.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 7.88e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 6.71e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 7.07e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.62e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.20e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 3.06e3T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.18e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.59e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.16e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.16e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 8.06e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.79e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 6.74e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 5.66e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.18e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.61e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.04e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.90e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 9.01e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 7.01e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.60e6T + 8.07e13T^{2} \) |
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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
−12.73669312029170055193720093257, −11.17602873727958656130726069858, −9.954107135507584338454003554118, −9.009485056307492176726600151792, −8.682966096656110961557473608937, −6.93663391244607833545606553645, −5.94658560770147008731835373543, −3.75758279021142527764007846891, −2.26240810564051360534339598441, −1.38329015496585012641442087136,
1.38329015496585012641442087136, 2.26240810564051360534339598441, 3.75758279021142527764007846891, 5.94658560770147008731835373543, 6.93663391244607833545606553645, 8.682966096656110961557473608937, 9.009485056307492176726600151792, 9.954107135507584338454003554118, 11.17602873727958656130726069858, 12.73669312029170055193720093257
