| L(s) = 1 | + 1.41·3-s + 2·5-s + 2.82·7-s − 0.999·9-s − 4.24·11-s + 6·13-s + 2.82·15-s + 4.24·19-s + 4.00·21-s − 8.48·23-s − 25-s − 5.65·27-s + 2·29-s − 5.65·31-s − 6·33-s + 5.65·35-s + 6·37-s + 8.48·39-s + 6·41-s + 4.24·43-s − 1.99·45-s + 1.00·49-s − 2·53-s − 8.48·55-s + 6·57-s − 1.41·59-s + 6·61-s + ⋯ |
| L(s) = 1 | + 0.816·3-s + 0.894·5-s + 1.06·7-s − 0.333·9-s − 1.27·11-s + 1.66·13-s + 0.730·15-s + 0.973·19-s + 0.872·21-s − 1.76·23-s − 0.200·25-s − 1.08·27-s + 0.371·29-s − 1.01·31-s − 1.04·33-s + 0.956·35-s + 0.986·37-s + 1.35·39-s + 0.937·41-s + 0.646·43-s − 0.298·45-s + 0.142·49-s − 0.274·53-s − 1.14·55-s + 0.794·57-s − 0.184·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.218250949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.218250949\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4.24T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 + 12T + 73T^{2} \) |
| 79 | \( 1 - 5.65T + 79T^{2} \) |
| 83 | \( 1 + 4.24T + 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−10.88820804870763015630312509868, −9.939496571832144260695678868519, −9.036800116186650427672448385071, −8.136209404913370480790908559261, −7.71500601574158399637037578135, −5.98859826938756647506654502941, −5.47761574613046253906233273467, −4.03141419798265746295869587755, −2.73263639610669961746742872748, −1.68277690453840397387089625321,
1.68277690453840397387089625321, 2.73263639610669961746742872748, 4.03141419798265746295869587755, 5.47761574613046253906233273467, 5.98859826938756647506654502941, 7.71500601574158399637037578135, 8.136209404913370480790908559261, 9.036800116186650427672448385071, 9.939496571832144260695678868519, 10.88820804870763015630312509868
