| L(s) = 1 | + 14.1·2-s + 72.7·4-s − 125·5-s − 1.58e3·7-s − 782.·8-s − 1.77e3·10-s − 5.44e3·11-s + 8.84e3·13-s − 2.24e4·14-s − 2.04e4·16-s − 2.85e4·17-s + 1.76e4·19-s − 9.09e3·20-s − 7.71e4·22-s − 3.19e4·23-s + 1.56e4·25-s + 1.25e5·26-s − 1.15e5·28-s + 6.15e4·29-s + 2.54e5·31-s − 1.88e5·32-s − 4.04e5·34-s + 1.98e5·35-s − 5.44e5·37-s + 2.49e5·38-s + 9.78e4·40-s + 1.03e5·41-s + ⋯ |
| L(s) = 1 | + 1.25·2-s + 0.568·4-s − 0.447·5-s − 1.74·7-s − 0.540·8-s − 0.560·10-s − 1.23·11-s + 1.11·13-s − 2.19·14-s − 1.24·16-s − 1.40·17-s + 0.588·19-s − 0.254·20-s − 1.54·22-s − 0.548·23-s + 0.199·25-s + 1.39·26-s − 0.994·28-s + 0.468·29-s + 1.53·31-s − 1.01·32-s − 1.76·34-s + 0.782·35-s − 1.76·37-s + 0.737·38-s + 0.241·40-s + 0.233·41-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
| \(L(4)\) |
\(\approx\) |
\(1.421370925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.421370925\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 14.1T + 128T^{2} \) |
| 7 | \( 1 + 1.58e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.44e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 8.84e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.85e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.76e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.19e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.15e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.54e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.44e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.03e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.78e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.00e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 3.73e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.22e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.60e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.56e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.98e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.69e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.34e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.13e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.71e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.66e6T + 8.07e13T^{2} \) |
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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
−10.20402674294434111569207314994, −9.180828790118648233353019878617, −8.238139553780049154226162773626, −6.79648464893178193757040113323, −6.25350507148110817331052830700, −5.22847910136540929378319690341, −4.11914042939635567785725968733, −3.30560930932901812036658044439, −2.54121769174082790350090336513, −0.42278495056528303849512721130,
0.42278495056528303849512721130, 2.54121769174082790350090336513, 3.30560930932901812036658044439, 4.11914042939635567785725968733, 5.22847910136540929378319690341, 6.25350507148110817331052830700, 6.79648464893178193757040113323, 8.238139553780049154226162773626, 9.180828790118648233353019878617, 10.20402674294434111569207314994
