| L(s) = 1 | + 14.5·2-s + 84.3·4-s + 404.·5-s − 387.·7-s − 636.·8-s + 5.89e3·10-s − 6.03e3·11-s + 1.43e4·13-s − 5.65e3·14-s − 2.00e4·16-s + 2.91e4·17-s + 3.30e4·19-s + 3.41e4·20-s − 8.79e4·22-s + 1.21e4·23-s + 8.57e4·25-s + 2.08e5·26-s − 3.27e4·28-s + 6.20e4·29-s + 1.78e5·31-s − 2.10e5·32-s + 4.25e5·34-s − 1.57e5·35-s + 1.43e4·37-s + 4.81e5·38-s − 2.57e5·40-s + 2.80e5·41-s + ⋯ |
| L(s) = 1 | + 1.28·2-s + 0.658·4-s + 1.44·5-s − 0.427·7-s − 0.439·8-s + 1.86·10-s − 1.36·11-s + 1.80·13-s − 0.550·14-s − 1.22·16-s + 1.44·17-s + 1.10·19-s + 0.954·20-s − 1.76·22-s + 0.208·23-s + 1.09·25-s + 2.32·26-s − 0.281·28-s + 0.472·29-s + 1.07·31-s − 1.13·32-s + 1.85·34-s − 0.619·35-s + 0.0464·37-s + 1.42·38-s − 0.636·40-s + 0.635·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.431082310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.431082310\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - 1.21e4T \) |
| good | 2 | \( 1 - 14.5T + 128T^{2} \) |
| 5 | \( 1 - 404.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 387.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 6.03e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.43e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.91e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.30e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 6.20e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.78e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.43e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.80e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.24e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.74e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.02e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 9.21e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 5.58e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.65e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.24e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.68e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.49e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.51e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.17e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.37e7T + 8.07e13T^{2} \) |
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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
−11.24895245767822522778630770774, −10.17082636296789264426560501406, −9.368894060135625430412891690674, −8.051033002067829099459018040096, −6.42019470671843925237076277242, −5.75271331787734722140709932478, −5.06626868648033263534117076383, −3.46902153786386229581058024885, −2.64780708787147335158195943874, −1.09142684682670622109101415157,
1.09142684682670622109101415157, 2.64780708787147335158195943874, 3.46902153786386229581058024885, 5.06626868648033263534117076383, 5.75271331787734722140709932478, 6.42019470671843925237076277242, 8.051033002067829099459018040096, 9.368894060135625430412891690674, 10.17082636296789264426560501406, 11.24895245767822522778630770774
