L(s) = 1 | + 8·5-s + 7·7-s − 40·11-s − 12·13-s + 58·17-s − 26·19-s − 64·23-s − 61·25-s + 62·29-s − 252·31-s + 56·35-s + 26·37-s − 6·41-s − 416·43-s − 396·47-s + 49·49-s + 450·53-s − 320·55-s + 274·59-s − 576·61-s − 96·65-s + 476·67-s − 448·71-s − 158·73-s − 280·77-s + 936·79-s + 530·83-s + ⋯ |
L(s) = 1 | + 0.715·5-s + 0.377·7-s − 1.09·11-s − 0.256·13-s + 0.827·17-s − 0.313·19-s − 0.580·23-s − 0.487·25-s + 0.397·29-s − 1.46·31-s + 0.270·35-s + 0.115·37-s − 0.0228·41-s − 1.47·43-s − 1.22·47-s + 1/7·49-s + 1.16·53-s − 0.784·55-s + 0.604·59-s − 1.20·61-s − 0.183·65-s + 0.867·67-s − 0.748·71-s − 0.253·73-s − 0.414·77-s + 1.33·79-s + 0.700·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
good | 5 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 40 T + p^{3} T^{2} \) |
| 13 | \( 1 + 12 T + p^{3} T^{2} \) |
| 17 | \( 1 - 58 T + p^{3} T^{2} \) |
| 19 | \( 1 + 26 T + p^{3} T^{2} \) |
| 23 | \( 1 + 64 T + p^{3} T^{2} \) |
| 29 | \( 1 - 62 T + p^{3} T^{2} \) |
| 31 | \( 1 + 252 T + p^{3} T^{2} \) |
| 37 | \( 1 - 26 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 T + p^{3} T^{2} \) |
| 43 | \( 1 + 416 T + p^{3} T^{2} \) |
| 47 | \( 1 + 396 T + p^{3} T^{2} \) |
| 53 | \( 1 - 450 T + p^{3} T^{2} \) |
| 59 | \( 1 - 274 T + p^{3} T^{2} \) |
| 61 | \( 1 + 576 T + p^{3} T^{2} \) |
| 67 | \( 1 - 476 T + p^{3} T^{2} \) |
| 71 | \( 1 + 448 T + p^{3} T^{2} \) |
| 73 | \( 1 + 158 T + p^{3} T^{2} \) |
| 79 | \( 1 - 936 T + p^{3} T^{2} \) |
| 83 | \( 1 - 530 T + p^{3} T^{2} \) |
| 89 | \( 1 - 390 T + p^{3} T^{2} \) |
| 97 | \( 1 - 214 T + p^{3} T^{2} \) |
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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
−9.279274704300547730523524788687, −8.216143852122074665633956411205, −7.64375968122612010146055755287, −6.55880109228085385189909556688, −5.56412989895551819632135340069, −5.02873362925692431004492977050, −3.72850842341197016567627414298, −2.53346841122228334214817690594, −1.59868074717443280620767194070, 0,
1.59868074717443280620767194070, 2.53346841122228334214817690594, 3.72850842341197016567627414298, 5.02873362925692431004492977050, 5.56412989895551819632135340069, 6.55880109228085385189909556688, 7.64375968122612010146055755287, 8.216143852122074665633956411205, 9.279274704300547730523524788687