| L(s) = 1 | − 2.44·2-s + 3.98·4-s − 0.269·7-s − 4.84·8-s − 5.96·11-s − 0.658·13-s + 0.658·14-s + 3.89·16-s + 5.29·17-s + 3.07·19-s + 14.5·22-s − 9.02·23-s + 1.61·26-s − 1.07·28-s + 4.49·29-s + 1.73·31-s + 0.176·32-s − 12.9·34-s + 37-s − 7.51·38-s − 7.16·41-s + 8.05·43-s − 23.7·44-s + 22.0·46-s − 8.62·47-s − 6.92·49-s − 2.62·52-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 1.99·4-s − 0.101·7-s − 1.71·8-s − 1.79·11-s − 0.182·13-s + 0.176·14-s + 0.972·16-s + 1.28·17-s + 0.704·19-s + 3.10·22-s − 1.88·23-s + 0.315·26-s − 0.202·28-s + 0.833·29-s + 0.310·31-s + 0.0312·32-s − 2.22·34-s + 0.164·37-s − 1.21·38-s − 1.11·41-s + 1.22·43-s − 3.57·44-s + 3.25·46-s − 1.25·47-s − 0.989·49-s − 0.363·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 7 | \( 1 + 0.269T + 7T^{2} \) |
| 11 | \( 1 + 5.96T + 11T^{2} \) |
| 13 | \( 1 + 0.658T + 13T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 19 | \( 1 - 3.07T + 19T^{2} \) |
| 23 | \( 1 + 9.02T + 23T^{2} \) |
| 29 | \( 1 - 4.49T + 29T^{2} \) |
| 31 | \( 1 - 1.73T + 31T^{2} \) |
| 41 | \( 1 + 7.16T + 41T^{2} \) |
| 43 | \( 1 - 8.05T + 43T^{2} \) |
| 47 | \( 1 + 8.62T + 47T^{2} \) |
| 53 | \( 1 - 5.81T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 - 7.78T + 61T^{2} \) |
| 67 | \( 1 - 0.269T + 67T^{2} \) |
| 71 | \( 1 + 3.30T + 71T^{2} \) |
| 73 | \( 1 - 3.69T + 73T^{2} \) |
| 79 | \( 1 - 8.85T + 79T^{2} \) |
| 83 | \( 1 + 5.34T + 83T^{2} \) |
| 89 | \( 1 + 5.20T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−7.68444034811525835633744624297, −7.17608126103048707722321573066, −6.26221004165501036519832339716, −5.56879328113231733941145809156, −4.84210159894273116909882050811, −3.56267448669435926049576808714, −2.70677750544045974759741662898, −2.06097226251891460312542605871, −0.962877783285697247670829600565, 0,
0.962877783285697247670829600565, 2.06097226251891460312542605871, 2.70677750544045974759741662898, 3.56267448669435926049576808714, 4.84210159894273116909882050811, 5.56879328113231733941145809156, 6.26221004165501036519832339716, 7.17608126103048707722321573066, 7.68444034811525835633744624297
