| L(s) = 1 | + 2.67·2-s − 3-s + 5.15·4-s − 2.67·6-s + 2.80·7-s + 8.44·8-s + 9-s − 11-s − 5.15·12-s − 5.11·13-s + 7.50·14-s + 12.2·16-s + 4.54·17-s + 2.67·18-s − 4.57·19-s − 2.80·21-s − 2.67·22-s + 4·23-s − 8.44·24-s − 13.6·26-s − 27-s + 14.4·28-s − 2.38·29-s − 0.962·31-s + 15.9·32-s + 33-s + 12.1·34-s + ⋯ |
| L(s) = 1 | + 1.89·2-s − 0.577·3-s + 2.57·4-s − 1.09·6-s + 1.06·7-s + 2.98·8-s + 0.333·9-s − 0.301·11-s − 1.48·12-s − 1.41·13-s + 2.00·14-s + 3.06·16-s + 1.10·17-s + 0.630·18-s − 1.04·19-s − 0.612·21-s − 0.570·22-s + 0.834·23-s − 1.72·24-s − 2.68·26-s − 0.192·27-s + 2.73·28-s − 0.443·29-s − 0.172·31-s + 2.81·32-s + 0.174·33-s + 2.08·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.370916874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.370916874\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 7 | \( 1 - 2.80T + 7T^{2} \) |
| 13 | \( 1 + 5.11T + 13T^{2} \) |
| 17 | \( 1 - 4.54T + 17T^{2} \) |
| 19 | \( 1 + 4.57T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 2.38T + 29T^{2} \) |
| 31 | \( 1 + 0.962T + 31T^{2} \) |
| 37 | \( 1 - 1.61T + 37T^{2} \) |
| 41 | \( 1 + 2.38T + 41T^{2} \) |
| 43 | \( 1 - 2.80T + 43T^{2} \) |
| 47 | \( 1 + 4.31T + 47T^{2} \) |
| 53 | \( 1 + 6.57T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 - 7.92T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 7.35T + 71T^{2} \) |
| 73 | \( 1 - 6.41T + 73T^{2} \) |
| 79 | \( 1 - 1.35T + 79T^{2} \) |
| 83 | \( 1 + 0.806T + 83T^{2} \) |
| 89 | \( 1 + 2.96T + 89T^{2} \) |
| 97 | \( 1 + 9.92T + 97T^{2} \) |
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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
−10.77539098589023463741862450922, −9.743114596036594428248933507774, −8.042429170907351404348630653450, −7.34550030273298068337234858901, −6.47542481197842590998376621368, −5.39471893138383905450016926700, −4.98651758517345396645537036177, −4.16631700215969944884114739038, −2.88147475175510917268599223343, −1.75765100409066135275359600174,
1.75765100409066135275359600174, 2.88147475175510917268599223343, 4.16631700215969944884114739038, 4.98651758517345396645537036177, 5.39471893138383905450016926700, 6.47542481197842590998376621368, 7.34550030273298068337234858901, 8.042429170907351404348630653450, 9.743114596036594428248933507774, 10.77539098589023463741862450922
