| L(s) = 1 | − 5·2-s + 3·3-s + 17·4-s − 15·6-s − 7·7-s − 45·8-s + 9·9-s + 12·11-s + 51·12-s − 30·13-s + 35·14-s + 89·16-s + 134·17-s − 45·18-s − 92·19-s − 21·21-s − 60·22-s − 112·23-s − 135·24-s + 150·26-s + 27·27-s − 119·28-s − 58·29-s − 224·31-s − 85·32-s + 36·33-s − 670·34-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 0.577·3-s + 17/8·4-s − 1.02·6-s − 0.377·7-s − 1.98·8-s + 1/3·9-s + 0.328·11-s + 1.22·12-s − 0.640·13-s + 0.668·14-s + 1.39·16-s + 1.91·17-s − 0.589·18-s − 1.11·19-s − 0.218·21-s − 0.581·22-s − 1.01·23-s − 1.14·24-s + 1.13·26-s + 0.192·27-s − 0.803·28-s − 0.371·29-s − 1.29·31-s − 0.469·32-s + 0.189·33-s − 3.37·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 2 | \( 1 + 5 T + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 30 T + p^{3} T^{2} \) |
| 17 | \( 1 - 134 T + p^{3} T^{2} \) |
| 19 | \( 1 + 92 T + p^{3} T^{2} \) |
| 23 | \( 1 + 112 T + p^{3} T^{2} \) |
| 29 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 31 | \( 1 + 224 T + p^{3} T^{2} \) |
| 37 | \( 1 - 146 T + p^{3} T^{2} \) |
| 41 | \( 1 - 18 T + p^{3} T^{2} \) |
| 43 | \( 1 + 340 T + p^{3} T^{2} \) |
| 47 | \( 1 + 208 T + p^{3} T^{2} \) |
| 53 | \( 1 - 754 T + p^{3} T^{2} \) |
| 59 | \( 1 - 380 T + p^{3} T^{2} \) |
| 61 | \( 1 - 718 T + p^{3} T^{2} \) |
| 67 | \( 1 + 412 T + p^{3} T^{2} \) |
| 71 | \( 1 + 960 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1066 T + p^{3} T^{2} \) |
| 79 | \( 1 - 896 T + p^{3} T^{2} \) |
| 83 | \( 1 + 436 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1038 T + p^{3} T^{2} \) |
| 97 | \( 1 - 702 T + p^{3} T^{2} \) |
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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
−9.983368295346209557661824139070, −9.132399212286618113140429485776, −8.321160141010515972730626520114, −7.59807718789956922340133809470, −6.82857802699613954646027029549, −5.67713598014526538330494140942, −3.82780937631214780337091798568, −2.55442903366647574425109957770, −1.44864955116820776980174775076, 0,
1.44864955116820776980174775076, 2.55442903366647574425109957770, 3.82780937631214780337091798568, 5.67713598014526538330494140942, 6.82857802699613954646027029549, 7.59807718789956922340133809470, 8.321160141010515972730626520114, 9.132399212286618113140429485776, 9.983368295346209557661824139070
