| L(s) = 1 | − 2-s + 4-s + 0.459·5-s − 4.76·7-s − 8-s − 0.459·10-s − 0.347·11-s + 1.18·13-s + 4.76·14-s + 16-s + 3.30·17-s + 3.47·19-s + 0.459·20-s + 0.347·22-s − 23-s − 4.78·25-s − 1.18·26-s − 4.76·28-s − 2.65·29-s + 7.47·31-s − 32-s − 3.30·34-s − 2.19·35-s − 1.82·37-s − 3.47·38-s − 0.459·40-s − 10.0·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.205·5-s − 1.80·7-s − 0.353·8-s − 0.145·10-s − 0.104·11-s + 0.327·13-s + 1.27·14-s + 0.250·16-s + 0.801·17-s + 0.796·19-s + 0.102·20-s + 0.0740·22-s − 0.208·23-s − 0.957·25-s − 0.231·26-s − 0.900·28-s − 0.492·29-s + 1.34·31-s − 0.176·32-s − 0.567·34-s − 0.370·35-s − 0.299·37-s − 0.563·38-s − 0.0727·40-s − 1.57·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 0.459T + 5T^{2} \) |
| 7 | \( 1 + 4.76T + 7T^{2} \) |
| 11 | \( 1 + 0.347T + 11T^{2} \) |
| 13 | \( 1 - 1.18T + 13T^{2} \) |
| 17 | \( 1 - 3.30T + 17T^{2} \) |
| 19 | \( 1 - 3.47T + 19T^{2} \) |
| 29 | \( 1 + 2.65T + 29T^{2} \) |
| 31 | \( 1 - 7.47T + 31T^{2} \) |
| 37 | \( 1 + 1.82T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 6.99T + 47T^{2} \) |
| 53 | \( 1 - 0.836T + 53T^{2} \) |
| 59 | \( 1 + 5.59T + 59T^{2} \) |
| 61 | \( 1 - 1.98T + 61T^{2} \) |
| 67 | \( 1 + 7.70T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 - 5.28T + 73T^{2} \) |
| 79 | \( 1 + 0.865T + 79T^{2} \) |
| 83 | \( 1 - 9.38T + 83T^{2} \) |
| 89 | \( 1 + 8.11T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
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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
−8.152999477247982942095016414571, −7.43059306340975939331231849447, −6.72849957715305826960619120236, −6.00984873396583731884237621006, −5.49773500991637799315291177195, −4.05400341035725893815142528609, −3.26086481180161877845384339387, −2.57643003223849021611521203294, −1.22496407538322382092713373183, 0,
1.22496407538322382092713373183, 2.57643003223849021611521203294, 3.26086481180161877845384339387, 4.05400341035725893815142528609, 5.49773500991637799315291177195, 6.00984873396583731884237621006, 6.72849957715305826960619120236, 7.43059306340975939331231849447, 8.152999477247982942095016414571
