| L(s) = 1 | + 4·7-s + 4·11-s + 13-s + 2·17-s + 4·19-s − 6·29-s − 10·37-s − 2·41-s − 4·43-s − 4·47-s + 9·49-s − 6·53-s − 4·59-s − 2·61-s − 4·67-s − 8·71-s + 2·73-s + 16·77-s − 16·79-s − 4·83-s − 10·89-s + 4·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 1.20·11-s + 0.277·13-s + 0.485·17-s + 0.917·19-s − 1.11·29-s − 1.64·37-s − 0.312·41-s − 0.609·43-s − 0.583·47-s + 9/7·49-s − 0.824·53-s − 0.520·59-s − 0.256·61-s − 0.488·67-s − 0.949·71-s + 0.234·73-s + 1.82·77-s − 1.80·79-s − 0.439·83-s − 1.05·89-s + 0.419·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.18975546297053, −13.73155556282950, −13.15025500031225, −12.51030148667156, −11.85719335046158, −11.71389575522657, −11.22201219430436, −10.72826361733660, −10.11203622148130, −9.583349656278238, −9.004613245589905, −8.582694266384493, −8.099027589020992, −7.463249948032578, −7.145633255560716, −6.464821755222023, −5.750469284360545, −5.377927141842632, −4.717580352258418, −4.297725825765015, −3.503776113918556, −3.182545212275821, −2.072546485578353, −1.479555124268390, −1.243793088872229, 0,
1.243793088872229, 1.479555124268390, 2.072546485578353, 3.182545212275821, 3.503776113918556, 4.297725825765015, 4.717580352258418, 5.377927141842632, 5.750469284360545, 6.464821755222023, 7.145633255560716, 7.463249948032578, 8.099027589020992, 8.582694266384493, 9.004613245589905, 9.583349656278238, 10.11203622148130, 10.72826361733660, 11.22201219430436, 11.71389575522657, 11.85719335046158, 12.51030148667156, 13.15025500031225, 13.73155556282950, 14.18975546297053
