| L(s) = 1 | − 3-s − 2.57·5-s − 0.722·7-s + 9-s − 1.16·11-s + 2.87·13-s + 2.57·15-s + 6.88·17-s + 0.0742·19-s + 0.722·21-s + 6.64·23-s + 1.61·25-s − 27-s − 8.36·29-s − 6.34·31-s + 1.16·33-s + 1.85·35-s − 4.23·37-s − 2.87·39-s + 1.08·41-s + 3.62·43-s − 2.57·45-s − 1.46·47-s − 6.47·49-s − 6.88·51-s + 9.12·53-s + 2.99·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.15·5-s − 0.273·7-s + 0.333·9-s − 0.350·11-s + 0.798·13-s + 0.664·15-s + 1.66·17-s + 0.0170·19-s + 0.157·21-s + 1.38·23-s + 0.323·25-s − 0.192·27-s − 1.55·29-s − 1.13·31-s + 0.202·33-s + 0.314·35-s − 0.696·37-s − 0.461·39-s + 0.168·41-s + 0.552·43-s − 0.383·45-s − 0.213·47-s − 0.925·49-s − 0.963·51-s + 1.25·53-s + 0.403·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.013528716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.013528716\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 503 | \( 1 + T \) |
| good | 5 | \( 1 + 2.57T + 5T^{2} \) |
| 7 | \( 1 + 0.722T + 7T^{2} \) |
| 11 | \( 1 + 1.16T + 11T^{2} \) |
| 13 | \( 1 - 2.87T + 13T^{2} \) |
| 17 | \( 1 - 6.88T + 17T^{2} \) |
| 19 | \( 1 - 0.0742T + 19T^{2} \) |
| 23 | \( 1 - 6.64T + 23T^{2} \) |
| 29 | \( 1 + 8.36T + 29T^{2} \) |
| 31 | \( 1 + 6.34T + 31T^{2} \) |
| 37 | \( 1 + 4.23T + 37T^{2} \) |
| 41 | \( 1 - 1.08T + 41T^{2} \) |
| 43 | \( 1 - 3.62T + 43T^{2} \) |
| 47 | \( 1 + 1.46T + 47T^{2} \) |
| 53 | \( 1 - 9.12T + 53T^{2} \) |
| 59 | \( 1 + 4.35T + 59T^{2} \) |
| 61 | \( 1 - 5.87T + 61T^{2} \) |
| 67 | \( 1 - 0.793T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 9.16T + 73T^{2} \) |
| 79 | \( 1 + 6.61T + 79T^{2} \) |
| 83 | \( 1 + 9.56T + 83T^{2} \) |
| 89 | \( 1 + 9.56T + 89T^{2} \) |
| 97 | \( 1 - 1.48T + 97T^{2} \) |
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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
−7.903766416111653577025615697009, −7.37895156258811623326060268295, −6.83644388677576789324201100988, −5.67147600003572098022877945889, −5.45157492344764787567385707875, −4.35831287118463714476223427231, −3.60819121444589188735497216284, −3.11809587177218490964869130429, −1.61380551175231004243264897047, −0.56266378842928157673989980928,
0.56266378842928157673989980928, 1.61380551175231004243264897047, 3.11809587177218490964869130429, 3.60819121444589188735497216284, 4.35831287118463714476223427231, 5.45157492344764787567385707875, 5.67147600003572098022877945889, 6.83644388677576789324201100988, 7.37895156258811623326060268295, 7.903766416111653577025615697009
