| L(s) = 1 | − 0.187·2-s − 81·3-s − 511.·4-s − 625·5-s + 15.2·6-s + 905.·7-s + 192.·8-s + 6.56e3·9-s + 117.·10-s + 8.62e4·11-s + 4.14e4·12-s − 7.58e4·13-s − 169.·14-s + 5.06e4·15-s + 2.62e5·16-s − 3.13e5·17-s − 1.23e3·18-s + 1.30e5·19-s + 3.19e5·20-s − 7.33e4·21-s − 1.61e4·22-s + 1.18e6·23-s − 1.55e4·24-s + 3.90e5·25-s + 1.42e4·26-s − 5.31e5·27-s − 4.63e5·28-s + ⋯ |
| L(s) = 1 | − 0.00829·2-s − 0.577·3-s − 0.999·4-s − 0.447·5-s + 0.00479·6-s + 0.142·7-s + 0.0165·8-s + 0.333·9-s + 0.00371·10-s + 1.77·11-s + 0.577·12-s − 0.737·13-s − 0.00118·14-s + 0.258·15-s + 0.999·16-s − 0.910·17-s − 0.00276·18-s + 0.229·19-s + 0.447·20-s − 0.0822·21-s − 0.0147·22-s + 0.883·23-s − 0.00957·24-s + 0.200·25-s + 0.00611·26-s − 0.192·27-s − 0.142·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.029614663\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.029614663\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 5 | \( 1 + 625T \) |
| 19 | \( 1 - 1.30e5T \) |
| good | 2 | \( 1 + 0.187T + 512T^{2} \) |
| 7 | \( 1 - 905.T + 4.03e7T^{2} \) |
| 11 | \( 1 - 8.62e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 7.58e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.13e5T + 1.18e11T^{2} \) |
| 23 | \( 1 - 1.18e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.53e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.53e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 3.79e5T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.21e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 6.90e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.52e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.29e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 3.84e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 3.41e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.11e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.42e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.08e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 6.74e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 9.70e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.48e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.09e9T + 7.60e17T^{2} \) |
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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
−10.18060345324400381235847386387, −9.218382945456632124844278952376, −8.577163173916957583189942201763, −7.24995761899208855682326982093, −6.39356522782019151134945969256, −5.04339397561928763573100514307, −4.38415527838771827873350480918, −3.37836375733409083133909016834, −1.53345614812898033216802698957, −0.50219025444356863564505029755,
0.50219025444356863564505029755, 1.53345614812898033216802698957, 3.37836375733409083133909016834, 4.38415527838771827873350480918, 5.04339397561928763573100514307, 6.39356522782019151134945969256, 7.24995761899208855682326982093, 8.577163173916957583189942201763, 9.218382945456632124844278952376, 10.18060345324400381235847386387
