| L(s) = 1 | + 37.7·2-s + 81·3-s + 910.·4-s − 2.68e3·5-s + 3.05e3·6-s + 8.71e3·7-s + 1.50e4·8-s + 6.56e3·9-s − 1.01e5·10-s − 2.96e4·11-s + 7.37e4·12-s − 6.29e4·13-s + 3.28e5·14-s − 2.17e5·15-s + 1.00e5·16-s + 5.15e5·17-s + 2.47e5·18-s + 1.65e5·19-s − 2.44e6·20-s + 7.06e5·21-s − 1.11e6·22-s + 2.59e6·23-s + 1.21e6·24-s + 5.25e6·25-s − 2.37e6·26-s + 5.31e5·27-s + 7.93e6·28-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 0.577·3-s + 1.77·4-s − 1.92·5-s + 0.962·6-s + 1.37·7-s + 1.29·8-s + 0.333·9-s − 3.20·10-s − 0.610·11-s + 1.02·12-s − 0.611·13-s + 2.28·14-s − 1.10·15-s + 0.382·16-s + 1.49·17-s + 0.555·18-s + 0.290·19-s − 3.41·20-s + 0.792·21-s − 1.01·22-s + 1.93·23-s + 0.748·24-s + 2.69·25-s − 1.01·26-s + 0.192·27-s + 2.44·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(6.557526921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.557526921\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 59 | \( 1 + 1.21e7T \) |
| good | 2 | \( 1 - 37.7T + 512T^{2} \) |
| 5 | \( 1 + 2.68e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 8.71e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.96e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 6.29e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 5.15e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.65e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.59e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.90e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.31e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.29e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 4.06e5T + 3.27e14T^{2} \) |
| 43 | \( 1 + 8.02e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.31e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.16e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 1.69e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.43e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.15e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.21e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.46e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.03e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.89e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 7.57e8T + 7.60e17T^{2} \) |
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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
−11.52493257367668602551877094879, −10.54007200058973495792599226918, −8.496037448285675862975882214418, −7.74696308290664821638454779023, −7.00982805277039702258510124813, −5.00085193546159333433857162649, −4.72690169310353293139502470494, −3.48379674037278427489121805077, −2.77894953020444488854707245761, −1.00690168959864242710996474705,
1.00690168959864242710996474705, 2.77894953020444488854707245761, 3.48379674037278427489121805077, 4.72690169310353293139502470494, 5.00085193546159333433857162649, 7.00982805277039702258510124813, 7.74696308290664821638454779023, 8.496037448285675862975882214418, 10.54007200058973495792599226918, 11.52493257367668602551877094879
