| L(s) = 1 | − 20.0·2-s + 274.·4-s − 125·5-s − 1.43e3·7-s − 2.94e3·8-s + 2.50e3·10-s + 1.40e3·11-s − 428.·13-s + 2.87e4·14-s + 2.38e4·16-s − 6.31e3·17-s − 1.72e4·19-s − 3.43e4·20-s − 2.81e4·22-s + 3.02e4·23-s + 1.56e4·25-s + 8.58e3·26-s − 3.93e5·28-s − 1.82e5·29-s − 1.66e5·31-s − 1.02e5·32-s + 1.26e5·34-s + 1.79e5·35-s − 6.12e5·37-s + 3.46e5·38-s + 3.67e5·40-s + 3.14e5·41-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 2.14·4-s − 0.447·5-s − 1.57·7-s − 2.03·8-s + 0.793·10-s + 0.318·11-s − 0.0540·13-s + 2.80·14-s + 1.45·16-s − 0.311·17-s − 0.577·19-s − 0.959·20-s − 0.564·22-s + 0.518·23-s + 0.199·25-s + 0.0958·26-s − 3.38·28-s − 1.38·29-s − 1.00·31-s − 0.552·32-s + 0.552·34-s + 0.706·35-s − 1.98·37-s + 1.02·38-s + 0.908·40-s + 0.711·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.08656575486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08656575486\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 125T \) |
| good | 2 | \( 1 + 20.0T + 128T^{2} \) |
| 7 | \( 1 + 1.43e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.40e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 428.T + 6.27e7T^{2} \) |
| 17 | \( 1 + 6.31e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.72e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.02e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.82e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.66e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 6.12e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.14e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.49e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 8.00e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.25e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.09e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.30e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.62e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.64e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.61e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.03e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 9.11e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.31e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 3.75e6T + 8.07e13T^{2} \) |
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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
−9.901721718750295345285638209015, −9.121085665792196719433160906219, −8.571849016543612126179966876289, −7.25110446169181859655794909366, −6.88113910902225453051418981762, −5.77981769260929038358881542748, −3.85060840527502205534321821234, −2.79726011033369634316948447151, −1.56070642186163523905807268576, −0.17549492053531014704780474015,
0.17549492053531014704780474015, 1.56070642186163523905807268576, 2.79726011033369634316948447151, 3.85060840527502205534321821234, 5.77981769260929038358881542748, 6.88113910902225453051418981762, 7.25110446169181859655794909366, 8.571849016543612126179966876289, 9.121085665792196719433160906219, 9.901721718750295345285638209015
