| L(s) = 1 | + 4·5-s − 7-s − 2·13-s + 4·17-s − 3·19-s − 2·23-s + 11·25-s − 6·29-s − 5·31-s − 4·35-s − 3·37-s − 2·41-s + 12·43-s − 2·47-s − 6·49-s + 6·53-s − 10·59-s + 3·61-s − 8·65-s + 67-s + 11·73-s − 11·79-s − 6·83-s + 16·85-s − 12·89-s + 2·91-s − 12·95-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.377·7-s − 0.554·13-s + 0.970·17-s − 0.688·19-s − 0.417·23-s + 11/5·25-s − 1.11·29-s − 0.898·31-s − 0.676·35-s − 0.493·37-s − 0.312·41-s + 1.82·43-s − 0.291·47-s − 6/7·49-s + 0.824·53-s − 1.30·59-s + 0.384·61-s − 0.992·65-s + 0.122·67-s + 1.28·73-s − 1.23·79-s − 0.658·83-s + 1.73·85-s − 1.27·89-s + 0.209·91-s − 1.23·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−14.42157016438538, −13.96894912945373, −13.32539688466106, −12.96701971316014, −12.48775958495328, −12.13840087598515, −11.17553380454838, −10.83715064628041, −10.19655669763757, −9.790495097078583, −9.490140410471795, −8.930412688453835, −8.398072803133767, −7.570578646353295, −7.143066676554933, −6.516662038340556, −5.839748204272500, −5.713888169873093, −5.067633101164371, −4.386061465733130, −3.579541845061764, −2.972414354199864, −2.218948353519293, −1.860367773705862, −1.076168670256503, 0,
1.076168670256503, 1.860367773705862, 2.218948353519293, 2.972414354199864, 3.579541845061764, 4.386061465733130, 5.067633101164371, 5.713888169873093, 5.839748204272500, 6.516662038340556, 7.143066676554933, 7.570578646353295, 8.398072803133767, 8.930412688453835, 9.490140410471795, 9.790495097078583, 10.19655669763757, 10.83715064628041, 11.17553380454838, 12.13840087598515, 12.48775958495328, 12.96701971316014, 13.32539688466106, 13.96894912945373, 14.42157016438538
