| L(s) = 1 | − 2-s + 4-s − 4.44·7-s − 8-s − 1.44·11-s + 2.44·13-s + 4.44·14-s + 16-s + 3.89·17-s − 0.550·19-s + 1.44·22-s − 2.89·23-s − 2.44·26-s − 4.44·28-s + 6·29-s + 6.44·31-s − 32-s − 3.89·34-s − 8·37-s + 0.550·38-s + 41-s − 7.44·43-s − 1.44·44-s + 2.89·46-s + 0.449·47-s + 12.7·49-s + 2.44·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.68·7-s − 0.353·8-s − 0.437·11-s + 0.679·13-s + 1.18·14-s + 0.250·16-s + 0.945·17-s − 0.126·19-s + 0.309·22-s − 0.604·23-s − 0.480·26-s − 0.840·28-s + 1.11·29-s + 1.15·31-s − 0.176·32-s − 0.668·34-s − 1.31·37-s + 0.0893·38-s + 0.156·41-s − 1.13·43-s − 0.218·44-s + 0.427·46-s + 0.0655·47-s + 1.82·49-s + 0.339·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4.44T + 7T^{2} \) |
| 11 | \( 1 + 1.44T + 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 - 3.89T + 17T^{2} \) |
| 19 | \( 1 + 0.550T + 19T^{2} \) |
| 23 | \( 1 + 2.89T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 6.44T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 - T + 41T^{2} \) |
| 43 | \( 1 + 7.44T + 43T^{2} \) |
| 47 | \( 1 - 0.449T + 47T^{2} \) |
| 53 | \( 1 - 8.44T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 + 0.449T + 61T^{2} \) |
| 67 | \( 1 + 9.44T + 67T^{2} \) |
| 71 | \( 1 + 2.44T + 71T^{2} \) |
| 73 | \( 1 - 4.79T + 73T^{2} \) |
| 79 | \( 1 + 7.34T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 + 13T + 97T^{2} \) |
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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
−8.302815122357674970102595899963, −7.29154537082269082549887394396, −6.65804367629973027316048428085, −6.07241967297884031258675785047, −5.31771719187244213030494224087, −4.01285725293355569658865985821, −3.23186992717430230942811359677, −2.56242775054916790578048754790, −1.17631391823707943161975143880, 0,
1.17631391823707943161975143880, 2.56242775054916790578048754790, 3.23186992717430230942811359677, 4.01285725293355569658865985821, 5.31771719187244213030494224087, 6.07241967297884031258675785047, 6.65804367629973027316048428085, 7.29154537082269082549887394396, 8.302815122357674970102595899963
