| L(s) = 1 | − 2-s + 4-s + 3.23·5-s − 7-s − 8-s − 3.23·10-s + 0.763·11-s + 1.23·13-s + 14-s + 16-s − 7.70·17-s + 19-s + 3.23·20-s − 0.763·22-s + 5.23·23-s + 5.47·25-s − 1.23·26-s − 28-s + 8.47·29-s − 2·31-s − 32-s + 7.70·34-s − 3.23·35-s − 10.4·37-s − 38-s − 3.23·40-s + 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.44·5-s − 0.377·7-s − 0.353·8-s − 1.02·10-s + 0.230·11-s + 0.342·13-s + 0.267·14-s + 0.250·16-s − 1.86·17-s + 0.229·19-s + 0.723·20-s − 0.162·22-s + 1.09·23-s + 1.09·25-s − 0.242·26-s − 0.188·28-s + 1.57·29-s − 0.359·31-s − 0.176·32-s + 1.32·34-s − 0.546·35-s − 1.72·37-s − 0.162·38-s − 0.511·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.665953368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.665953368\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 11 | \( 1 - 0.763T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 + 7.70T + 17T^{2} \) |
| 23 | \( 1 - 5.23T + 23T^{2} \) |
| 29 | \( 1 - 8.47T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 8.94T + 43T^{2} \) |
| 47 | \( 1 - 8.94T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 0.472T + 61T^{2} \) |
| 67 | \( 1 - 0.291T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 0.763T + 79T^{2} \) |
| 83 | \( 1 + 10T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 4.76T + 97T^{2} \) |
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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
−8.858885999448037013391045226848, −8.715010433521129373296102434571, −7.22325951581538230390779888314, −6.72237546101772004308449263692, −6.01899377163174968850444715338, −5.24405013490874731874129327005, −4.12144236108976087326420574675, −2.77473240337934365991383204539, −2.11753775817860732627324061317, −0.941750525556653715752685405721,
0.941750525556653715752685405721, 2.11753775817860732627324061317, 2.77473240337934365991383204539, 4.12144236108976087326420574675, 5.24405013490874731874129327005, 6.01899377163174968850444715338, 6.72237546101772004308449263692, 7.22325951581538230390779888314, 8.715010433521129373296102434571, 8.858885999448037013391045226848
