| L(s) = 1 | + 3-s − 4·7-s + 9-s − 3·11-s − 2·13-s − 8·19-s − 4·21-s + 6·23-s + 27-s + 3·29-s + 7·31-s − 3·33-s − 8·37-s − 2·39-s + 6·41-s − 4·43-s − 6·47-s + 9·49-s + 9·53-s − 8·57-s − 15·59-s + 14·61-s − 4·63-s + 2·67-s + 6·69-s + 7·73-s + 12·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.904·11-s − 0.554·13-s − 1.83·19-s − 0.872·21-s + 1.25·23-s + 0.192·27-s + 0.557·29-s + 1.25·31-s − 0.522·33-s − 1.31·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.875·47-s + 9/7·49-s + 1.23·53-s − 1.05·57-s − 1.95·59-s + 1.79·61-s − 0.503·63-s + 0.244·67-s + 0.722·69-s + 0.819·73-s + 1.36·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 346800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 346800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.402056151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.402056151\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.78289040240516, −12.16167568132696, −11.88932996494101, −10.93772123561668, −10.70264552645748, −10.19765114357770, −9.893612965956972, −9.370492526014778, −8.913616046353262, −8.447211920136566, −8.093800819950048, −7.451923896870021, −6.896300030161702, −6.617254319992234, −6.201466030017745, −5.540335877120027, −4.890493178481707, −4.574227735926771, −3.874093112427404, −3.297569164772021, −2.960111517444160, −2.356887518760824, −2.042563439236489, −0.9583381249536497, −0.3324062726067401,
0.3324062726067401, 0.9583381249536497, 2.042563439236489, 2.356887518760824, 2.960111517444160, 3.297569164772021, 3.874093112427404, 4.574227735926771, 4.890493178481707, 5.540335877120027, 6.201466030017745, 6.617254319992234, 6.896300030161702, 7.451923896870021, 8.093800819950048, 8.447211920136566, 8.913616046353262, 9.370492526014778, 9.893612965956972, 10.19765114357770, 10.70264552645748, 10.93772123561668, 11.88932996494101, 12.16167568132696, 12.78289040240516
