| L(s) = 1 | − 5-s + 2.56·7-s − 1.43·11-s + 13-s − 5.68·17-s + 5.12·19-s − 1.43·23-s + 25-s + 2·29-s − 1.12·31-s − 2.56·35-s − 10.8·37-s + 9.68·41-s − 6.24·43-s − 1.12·47-s − 0.438·49-s + 0.561·53-s + 1.43·55-s − 8·59-s + 1.68·61-s − 65-s − 2.24·67-s − 7.68·71-s − 0.246·73-s − 3.68·77-s + 8.80·79-s − 8·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.968·7-s − 0.433·11-s + 0.277·13-s − 1.37·17-s + 1.17·19-s − 0.299·23-s + 0.200·25-s + 0.371·29-s − 0.201·31-s − 0.432·35-s − 1.77·37-s + 1.51·41-s − 0.952·43-s − 0.163·47-s − 0.0626·49-s + 0.0771·53-s + 0.193·55-s − 1.04·59-s + 0.215·61-s − 0.124·65-s − 0.274·67-s − 0.912·71-s − 0.0288·73-s − 0.419·77-s + 0.990·79-s − 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 + 1.43T + 11T^{2} \) |
| 17 | \( 1 + 5.68T + 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 23 | \( 1 + 1.43T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 1.12T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 - 9.68T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 + 1.12T + 47T^{2} \) |
| 53 | \( 1 - 0.561T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 1.68T + 61T^{2} \) |
| 67 | \( 1 + 2.24T + 67T^{2} \) |
| 71 | \( 1 + 7.68T + 71T^{2} \) |
| 73 | \( 1 + 0.246T + 73T^{2} \) |
| 79 | \( 1 - 8.80T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 - 2.31T + 89T^{2} \) |
| 97 | \( 1 - 2.80T + 97T^{2} \) |
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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
−7.46826703227018108347205052058, −6.79047534746052552594223340169, −5.98136787236832127037925946588, −5.12293989773460976487171809272, −4.69193518259307691370105208448, −3.88101806253099510552626210830, −3.06012365866074770359348425611, −2.12866735027996541693964132644, −1.27851577796782012337582831345, 0,
1.27851577796782012337582831345, 2.12866735027996541693964132644, 3.06012365866074770359348425611, 3.88101806253099510552626210830, 4.69193518259307691370105208448, 5.12293989773460976487171809272, 5.98136787236832127037925946588, 6.79047534746052552594223340169, 7.46826703227018108347205052058
