| L(s) = 1 | − 2.68·2-s + 5.20·4-s − 3.48·7-s − 8.61·8-s + 3.18·11-s − 1.81·13-s + 9.35·14-s + 12.7·16-s − 2.82·17-s − 5.72·19-s − 8.56·22-s + 1.43·23-s + 4.87·26-s − 18.1·28-s + 2.30·29-s + 6.00·31-s − 16.9·32-s + 7.58·34-s − 37-s + 15.3·38-s − 7.82·41-s + 11.6·43-s + 16.6·44-s − 3.84·46-s − 2.42·47-s + 5.13·49-s − 9.46·52-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 2.60·4-s − 1.31·7-s − 3.04·8-s + 0.961·11-s − 0.503·13-s + 2.49·14-s + 3.17·16-s − 0.685·17-s − 1.31·19-s − 1.82·22-s + 0.298·23-s + 0.956·26-s − 3.42·28-s + 0.428·29-s + 1.07·31-s − 2.98·32-s + 1.30·34-s − 0.164·37-s + 2.49·38-s − 1.22·41-s + 1.78·43-s + 2.50·44-s − 0.567·46-s − 0.354·47-s + 0.732·49-s − 1.31·52-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 | \( 1 + 2.68T + 2T^{2} \) |
| 7 | \( 1 + 3.48T + 7T^{2} \) |
| 11 | \( 1 - 3.18T + 11T^{2} \) |
| 13 | \( 1 + 1.81T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + 5.72T + 19T^{2} \) |
| 23 | \( 1 - 1.43T + 23T^{2} \) |
| 29 | \( 1 - 2.30T + 29T^{2} \) |
| 31 | \( 1 - 6.00T + 31T^{2} \) |
| 41 | \( 1 + 7.82T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 + 2.42T + 47T^{2} \) |
| 53 | \( 1 - 9.32T + 53T^{2} \) |
| 59 | \( 1 + 2.86T + 59T^{2} \) |
| 61 | \( 1 + 4.67T + 61T^{2} \) |
| 67 | \( 1 - 6.00T + 67T^{2} \) |
| 71 | \( 1 + 3.40T + 71T^{2} \) |
| 73 | \( 1 - 1.89T + 73T^{2} \) |
| 79 | \( 1 + 1.60T + 79T^{2} \) |
| 83 | \( 1 + 7.37T + 83T^{2} \) |
| 89 | \( 1 - 8.48T + 89T^{2} \) |
| 97 | \( 1 - 9.00T + 97T^{2} \) |
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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
−7.49561185523055955361323132372, −6.83189260477514513107755967332, −6.46359238526872032011876169903, −5.94814765360023545180789802417, −4.56415911380884576029545013310, −3.57154734919857814788126906558, −2.70533778052655227887904850071, −2.05258319529922631645663924431, −0.918272729024876785292568136123, 0,
0.918272729024876785292568136123, 2.05258319529922631645663924431, 2.70533778052655227887904850071, 3.57154734919857814788126906558, 4.56415911380884576029545013310, 5.94814765360023545180789802417, 6.46359238526872032011876169903, 6.83189260477514513107755967332, 7.49561185523055955361323132372
