| L(s) = 1 | + 3-s − 3.23·5-s + 7-s − 2·9-s − 4.47·11-s + 0.236·13-s − 3.23·15-s + 7.23·19-s + 21-s + 23-s + 5.47·25-s − 5·27-s − 1.47·29-s + 9·31-s − 4.47·33-s − 3.23·35-s − 5.70·37-s + 0.236·39-s − 2.23·41-s − 2.47·43-s + 6.47·45-s + 3.47·47-s + 49-s + 11.2·53-s + 14.4·55-s + 7.23·57-s + 1.52·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.44·5-s + 0.377·7-s − 0.666·9-s − 1.34·11-s + 0.0654·13-s − 0.835·15-s + 1.66·19-s + 0.218·21-s + 0.208·23-s + 1.09·25-s − 0.962·27-s − 0.273·29-s + 1.61·31-s − 0.778·33-s − 0.546·35-s − 0.938·37-s + 0.0378·39-s − 0.349·41-s − 0.376·43-s + 0.964·45-s + 0.506·47-s + 0.142·49-s + 1.54·53-s + 1.95·55-s + 0.958·57-s + 0.198·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.328882350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.328882350\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 + 3.23T + 5T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 13 | \( 1 - 0.236T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 29 | \( 1 + 1.47T + 29T^{2} \) |
| 31 | \( 1 - 9T + 31T^{2} \) |
| 37 | \( 1 + 5.70T + 37T^{2} \) |
| 41 | \( 1 + 2.23T + 41T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 - 3.47T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 1.52T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 6.70T + 73T^{2} \) |
| 79 | \( 1 - 7.23T + 79T^{2} \) |
| 83 | \( 1 + 6.47T + 83T^{2} \) |
| 89 | \( 1 - 8.94T + 89T^{2} \) |
| 97 | \( 1 - 3.70T + 97T^{2} \) |
| 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
−8.489015612259476422521397059972, −8.212686644770663944643367150137, −7.57936137611453009807074963420, −6.91424316748024625755178213046, −5.51078413792474492063838923012, −5.01657147455405927444939910530, −3.89281773443994275913875936624, −3.19631257719102555392379570398, −2.41000161826508758512812095433, −0.68626938944931894447992867781,
0.68626938944931894447992867781, 2.41000161826508758512812095433, 3.19631257719102555392379570398, 3.89281773443994275913875936624, 5.01657147455405927444939910530, 5.51078413792474492063838923012, 6.91424316748024625755178213046, 7.57936137611453009807074963420, 8.212686644770663944643367150137, 8.489015612259476422521397059972
