| L(s) = 1 | + 8·2-s + 64·4-s + 206.·5-s + 1.61e3·7-s + 512·8-s + 1.65e3·10-s − 4.67e3·11-s + 6.67e3·13-s + 1.29e4·14-s + 4.09e3·16-s + 2.57e4·17-s + 2.23e4·19-s + 1.32e4·20-s − 3.74e4·22-s − 2.34e4·23-s − 3.54e4·25-s + 5.33e4·26-s + 1.03e5·28-s − 1.62e5·29-s − 2.33e5·31-s + 3.27e4·32-s + 2.05e5·34-s + 3.33e5·35-s + 3.08e5·37-s + 1.79e5·38-s + 1.05e5·40-s + 3.15e5·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.738·5-s + 1.78·7-s + 0.353·8-s + 0.522·10-s − 1.05·11-s + 0.842·13-s + 1.26·14-s + 0.250·16-s + 1.27·17-s + 0.748·19-s + 0.369·20-s − 0.749·22-s − 0.401·23-s − 0.454·25-s + 0.595·26-s + 0.891·28-s − 1.24·29-s − 1.40·31-s + 0.176·32-s + 0.898·34-s + 1.31·35-s + 1.00·37-s + 0.529·38-s + 0.261·40-s + 0.714·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.861565711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.861565711\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 206.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.61e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.67e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 6.67e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.57e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.23e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.34e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.62e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.33e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.08e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.15e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.22e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.95e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.69e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.12e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 9.03e4T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.52e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 7.03e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.06e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.67e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.77e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.07e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.44e6T + 8.07e13T^{2} \) |
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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
−11.45625265218072609229038225854, −10.84990202006842395661039271568, −9.674054447753008433786722522795, −8.166890914247731345078894499430, −7.48465265970786579254990993306, −5.65482901843056246993622209409, −5.31756461534359126466423811432, −3.85338187807361020576960303112, −2.26695216279643370105414153775, −1.27047633987558545339137356472,
1.27047633987558545339137356472, 2.26695216279643370105414153775, 3.85338187807361020576960303112, 5.31756461534359126466423811432, 5.65482901843056246993622209409, 7.48465265970786579254990993306, 8.166890914247731345078894499430, 9.674054447753008433786722522795, 10.84990202006842395661039271568, 11.45625265218072609229038225854
