| L(s) = 1 | + 1.16·2-s − 0.653·4-s + 5-s − 1.26·7-s − 3.07·8-s + 1.16·10-s + 2.81·11-s + 13-s − 1.46·14-s − 2.26·16-s − 1.42·17-s + 6.05·19-s − 0.653·20-s + 3.26·22-s + 2.77·23-s + 25-s + 1.16·26-s + 0.827·28-s − 1.23·29-s + 0.653·31-s + 3.53·32-s − 1.65·34-s − 1.26·35-s + 4.39·37-s + 7.02·38-s − 3.07·40-s + 2.50·41-s + ⋯ |
| L(s) = 1 | + 0.820·2-s − 0.326·4-s + 0.447·5-s − 0.478·7-s − 1.08·8-s + 0.366·10-s + 0.848·11-s + 0.277·13-s − 0.392·14-s − 0.566·16-s − 0.345·17-s + 1.38·19-s − 0.146·20-s + 0.696·22-s + 0.577·23-s + 0.200·25-s + 0.227·26-s + 0.156·28-s − 0.230·29-s + 0.117·31-s + 0.624·32-s − 0.283·34-s − 0.213·35-s + 0.723·37-s + 1.13·38-s − 0.486·40-s + 0.391·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.399546177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.399546177\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 1.16T + 2T^{2} \) |
| 7 | \( 1 + 1.26T + 7T^{2} \) |
| 11 | \( 1 - 2.81T + 11T^{2} \) |
| 17 | \( 1 + 1.42T + 17T^{2} \) |
| 19 | \( 1 - 6.05T + 19T^{2} \) |
| 23 | \( 1 - 2.77T + 23T^{2} \) |
| 29 | \( 1 + 1.23T + 29T^{2} \) |
| 31 | \( 1 - 0.653T + 31T^{2} \) |
| 37 | \( 1 - 4.39T + 37T^{2} \) |
| 41 | \( 1 - 2.50T + 41T^{2} \) |
| 43 | \( 1 - 6.27T + 43T^{2} \) |
| 47 | \( 1 - 9.88T + 47T^{2} \) |
| 53 | \( 1 - 8.18T + 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 + 0.957T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 - 9.63T + 71T^{2} \) |
| 73 | \( 1 + 5.09T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 - 1.43T + 83T^{2} \) |
| 89 | \( 1 - 5.90T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
| 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
−9.295112622194315438752342262885, −8.789200526230887459936827207502, −7.59465027791611769282004527279, −6.64996854292008635532632039242, −5.94922312467456246573478675001, −5.24950051882920372333857979296, −4.28635347094265909113868885186, −3.51399702097523467363112051783, −2.60235927690283174585784994962, −0.985922419222746993783635355478,
0.985922419222746993783635355478, 2.60235927690283174585784994962, 3.51399702097523467363112051783, 4.28635347094265909113868885186, 5.24950051882920372333857979296, 5.94922312467456246573478675001, 6.64996854292008635532632039242, 7.59465027791611769282004527279, 8.789200526230887459936827207502, 9.295112622194315438752342262885
