| L(s) = 1 | + 1.23·5-s − 3.23·7-s − 11-s + 5.23·13-s + 4.47·17-s − 6.47·19-s − 5.70·23-s − 3.47·25-s − 8.47·29-s − 4.00·35-s + 6·37-s − 10.9·41-s + 12.9·43-s + 5.70·47-s + 3.47·49-s − 6.76·53-s − 1.23·55-s − 5.52·59-s + 11.7·61-s + 6.47·65-s − 8·67-s − 5.70·71-s − 6·73-s + 3.23·77-s − 1.70·79-s + 4.94·83-s + 5.52·85-s + ⋯ |
| L(s) = 1 | + 0.552·5-s − 1.22·7-s − 0.301·11-s + 1.45·13-s + 1.08·17-s − 1.48·19-s − 1.19·23-s − 0.694·25-s − 1.57·29-s − 0.676·35-s + 0.986·37-s − 1.70·41-s + 1.97·43-s + 0.832·47-s + 0.496·49-s − 0.929·53-s − 0.166·55-s − 0.719·59-s + 1.49·61-s + 0.802·65-s − 0.977·67-s − 0.677·71-s − 0.702·73-s + 0.368·77-s − 0.192·79-s + 0.542·83-s + 0.599·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 + 3.23T + 7T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 + 5.70T + 23T^{2} \) |
| 29 | \( 1 + 8.47T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 12.9T + 43T^{2} \) |
| 47 | \( 1 - 5.70T + 47T^{2} \) |
| 53 | \( 1 + 6.76T + 53T^{2} \) |
| 59 | \( 1 + 5.52T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 5.70T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 1.70T + 79T^{2} \) |
| 83 | \( 1 - 4.94T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 8.47T + 97T^{2} \) |
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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
−8.310636520298814104968946354328, −7.62214456678762961095831972245, −6.58158715099616535878397542704, −5.95912243412198332862939200649, −5.64181583575188349998027152782, −4.13226813634703388643001432307, −3.62365314171810583063380223557, −2.57630195079804006875441651275, −1.53933414510189078224232579073, 0,
1.53933414510189078224232579073, 2.57630195079804006875441651275, 3.62365314171810583063380223557, 4.13226813634703388643001432307, 5.64181583575188349998027152782, 5.95912243412198332862939200649, 6.58158715099616535878397542704, 7.62214456678762961095831972245, 8.310636520298814104968946354328
