| L(s) = 1 | − 1.77·2-s + 1.16·4-s + 2.84·7-s + 1.49·8-s − 5.35·11-s + 2.50·13-s − 5.06·14-s − 4.97·16-s − 6.50·17-s + 2.25·19-s + 9.52·22-s + 0.355·23-s − 4.45·26-s + 3.31·28-s + 29-s + 5.45·31-s + 5.86·32-s + 11.5·34-s + 2.00·37-s − 4.00·38-s + 5.37·41-s − 12.8·43-s − 6.22·44-s − 0.632·46-s + 4.01·47-s + 1.11·49-s + 2.91·52-s + ⋯ |
| L(s) = 1 | − 1.25·2-s + 0.580·4-s + 1.07·7-s + 0.526·8-s − 1.61·11-s + 0.695·13-s − 1.35·14-s − 1.24·16-s − 1.57·17-s + 0.517·19-s + 2.03·22-s + 0.0742·23-s − 0.874·26-s + 0.625·28-s + 0.185·29-s + 0.980·31-s + 1.03·32-s + 1.98·34-s + 0.329·37-s − 0.650·38-s + 0.839·41-s − 1.96·43-s − 0.938·44-s − 0.0933·46-s + 0.585·47-s + 0.159·49-s + 0.403·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.77T + 2T^{2} \) |
| 7 | \( 1 - 2.84T + 7T^{2} \) |
| 11 | \( 1 + 5.35T + 11T^{2} \) |
| 13 | \( 1 - 2.50T + 13T^{2} \) |
| 17 | \( 1 + 6.50T + 17T^{2} \) |
| 19 | \( 1 - 2.25T + 19T^{2} \) |
| 23 | \( 1 - 0.355T + 23T^{2} \) |
| 31 | \( 1 - 5.45T + 31T^{2} \) |
| 37 | \( 1 - 2.00T + 37T^{2} \) |
| 41 | \( 1 - 5.37T + 41T^{2} \) |
| 43 | \( 1 + 12.8T + 43T^{2} \) |
| 47 | \( 1 - 4.01T + 47T^{2} \) |
| 53 | \( 1 - 3.42T + 53T^{2} \) |
| 59 | \( 1 - 4.76T + 59T^{2} \) |
| 61 | \( 1 + 4.29T + 61T^{2} \) |
| 67 | \( 1 - 1.09T + 67T^{2} \) |
| 71 | \( 1 + 5.21T + 71T^{2} \) |
| 73 | \( 1 - 6.78T + 73T^{2} \) |
| 79 | \( 1 + 1.84T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 - 2.56T + 89T^{2} \) |
| 97 | \( 1 + 2.73T + 97T^{2} \) |
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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
−7.971275094984447738324369370344, −7.23375755259683566669691040359, −6.51356426228232547386768188690, −5.45506748597904478943293718413, −4.80447868795655778519996817240, −4.18669783575102175977498377399, −2.80933060172415287717517533401, −2.06484410897063522977938067114, −1.14438197433490793176641561283, 0,
1.14438197433490793176641561283, 2.06484410897063522977938067114, 2.80933060172415287717517533401, 4.18669783575102175977498377399, 4.80447868795655778519996817240, 5.45506748597904478943293718413, 6.51356426228232547386768188690, 7.23375755259683566669691040359, 7.971275094984447738324369370344
