| L(s) = 1 | + 2.66·2-s − 0.205·3-s + 5.12·4-s + 2.58·5-s − 0.548·6-s − 2.90·7-s + 8.34·8-s − 2.95·9-s + 6.89·10-s − 1.27·11-s − 1.05·12-s − 0.244·13-s − 7.76·14-s − 0.530·15-s + 12.0·16-s + 4.82·17-s − 7.89·18-s − 6.43·19-s + 13.2·20-s + 0.597·21-s − 3.39·22-s − 5.37·23-s − 1.71·24-s + 1.67·25-s − 0.653·26-s + 1.22·27-s − 14.9·28-s + ⋯ |
| L(s) = 1 | + 1.88·2-s − 0.118·3-s + 2.56·4-s + 1.15·5-s − 0.223·6-s − 1.09·7-s + 2.95·8-s − 0.985·9-s + 2.18·10-s − 0.383·11-s − 0.303·12-s − 0.0678·13-s − 2.07·14-s − 0.136·15-s + 3.00·16-s + 1.17·17-s − 1.86·18-s − 1.47·19-s + 2.96·20-s + 0.130·21-s − 0.724·22-s − 1.12·23-s − 0.349·24-s + 0.335·25-s − 0.128·26-s + 0.235·27-s − 2.81·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.144946053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.144946053\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 503 | \( 1 - T \) |
| good | 2 | \( 1 - 2.66T + 2T^{2} \) |
| 3 | \( 1 + 0.205T + 3T^{2} \) |
| 5 | \( 1 - 2.58T + 5T^{2} \) |
| 7 | \( 1 + 2.90T + 7T^{2} \) |
| 11 | \( 1 + 1.27T + 11T^{2} \) |
| 13 | \( 1 + 0.244T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 + 6.43T + 19T^{2} \) |
| 23 | \( 1 + 5.37T + 23T^{2} \) |
| 29 | \( 1 - 5.00T + 29T^{2} \) |
| 31 | \( 1 + 5.77T + 31T^{2} \) |
| 37 | \( 1 + 0.0850T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 + 1.31T + 47T^{2} \) |
| 53 | \( 1 + 3.86T + 53T^{2} \) |
| 59 | \( 1 - 6.35T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 + 2.67T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 + 6.86T + 73T^{2} \) |
| 79 | \( 1 - 4.87T + 79T^{2} \) |
| 83 | \( 1 - 0.331T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−11.05358455761084032178705213667, −10.37054671923705472473825578631, −9.406662207479207183961468693796, −7.904829398927296031845625513825, −6.60859715386271174831768204057, −5.92070569704299324744657669235, −5.56469635819681823941105829890, −4.19171807737924744543727185110, −3.03075759241010325268939251295, −2.22408289099702569573792811229,
2.22408289099702569573792811229, 3.03075759241010325268939251295, 4.19171807737924744543727185110, 5.56469635819681823941105829890, 5.92070569704299324744657669235, 6.60859715386271174831768204057, 7.904829398927296031845625513825, 9.406662207479207183961468693796, 10.37054671923705472473825578631, 11.05358455761084032178705213667
