| L(s) = 1 | + 1.89·2-s − 3-s + 1.57·4-s − 5-s − 1.89·6-s − 4.66·7-s − 0.801·8-s + 9-s − 1.89·10-s − 1.18·11-s − 1.57·12-s − 13-s − 8.81·14-s + 15-s − 4.66·16-s − 3.45·17-s + 1.89·18-s − 6.98·19-s − 1.57·20-s + 4.66·21-s − 2.24·22-s + 3.07·23-s + 0.801·24-s + 25-s − 1.89·26-s − 27-s − 7.35·28-s + ⋯ |
| L(s) = 1 | + 1.33·2-s − 0.577·3-s + 0.788·4-s − 0.447·5-s − 0.772·6-s − 1.76·7-s − 0.283·8-s + 0.333·9-s − 0.598·10-s − 0.358·11-s − 0.455·12-s − 0.277·13-s − 2.35·14-s + 0.258·15-s − 1.16·16-s − 0.839·17-s + 0.445·18-s − 1.60·19-s − 0.352·20-s + 1.01·21-s − 0.479·22-s + 0.642·23-s + 0.163·24-s + 0.200·25-s − 0.370·26-s − 0.192·27-s − 1.38·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9333192248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9333192248\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 1.89T + 2T^{2} \) |
| 7 | \( 1 + 4.66T + 7T^{2} \) |
| 11 | \( 1 + 1.18T + 11T^{2} \) |
| 17 | \( 1 + 3.45T + 17T^{2} \) |
| 19 | \( 1 + 6.98T + 19T^{2} \) |
| 23 | \( 1 - 3.07T + 23T^{2} \) |
| 29 | \( 1 + 6.06T + 29T^{2} \) |
| 37 | \( 1 - 8.79T + 37T^{2} \) |
| 41 | \( 1 - 8.93T + 41T^{2} \) |
| 43 | \( 1 - 4.12T + 43T^{2} \) |
| 47 | \( 1 - 1.75T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 - 5.71T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 + 7.50T + 67T^{2} \) |
| 71 | \( 1 + 6.83T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 9.72T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 - 6.71T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−7.79836267736703197943854397873, −6.97271002738258428765648867939, −6.37471898843223418696353233748, −6.01928418755087623043020148389, −5.17323515256339453731152092168, −4.25658287161109148111260695758, −3.96269469844731751570097573661, −2.94174430905803538614186381363, −2.36512160037752548513457994124, −0.39266143839998015510239297023,
0.39266143839998015510239297023, 2.36512160037752548513457994124, 2.94174430905803538614186381363, 3.96269469844731751570097573661, 4.25658287161109148111260695758, 5.17323515256339453731152092168, 6.01928418755087623043020148389, 6.37471898843223418696353233748, 6.97271002738258428765648867939, 7.79836267736703197943854397873
