| L(s) = 1 | + 2-s − 0.235·3-s + 4-s + 1.96·5-s − 0.235·6-s + 7-s + 8-s − 2.94·9-s + 1.96·10-s − 1.72·11-s − 0.235·12-s − 2.88·13-s + 14-s − 0.463·15-s + 16-s + 4.64·17-s − 2.94·18-s − 0.307·19-s + 1.96·20-s − 0.235·21-s − 1.72·22-s − 7.97·23-s − 0.235·24-s − 1.12·25-s − 2.88·26-s + 1.39·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.135·3-s + 0.5·4-s + 0.879·5-s − 0.0961·6-s + 0.377·7-s + 0.353·8-s − 0.981·9-s + 0.622·10-s − 0.520·11-s − 0.0679·12-s − 0.799·13-s + 0.267·14-s − 0.119·15-s + 0.250·16-s + 1.12·17-s − 0.694·18-s − 0.0704·19-s + 0.439·20-s − 0.0513·21-s − 0.367·22-s − 1.66·23-s − 0.0480·24-s − 0.225·25-s − 0.565·26-s + 0.269·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 0.235T + 3T^{2} \) |
| 5 | \( 1 - 1.96T + 5T^{2} \) |
| 11 | \( 1 + 1.72T + 11T^{2} \) |
| 13 | \( 1 + 2.88T + 13T^{2} \) |
| 17 | \( 1 - 4.64T + 17T^{2} \) |
| 19 | \( 1 + 0.307T + 19T^{2} \) |
| 23 | \( 1 + 7.97T + 23T^{2} \) |
| 29 | \( 1 + 5.30T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 - 4.09T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 6.28T + 43T^{2} \) |
| 47 | \( 1 + 0.831T + 47T^{2} \) |
| 53 | \( 1 + 5.46T + 53T^{2} \) |
| 59 | \( 1 + 4.70T + 59T^{2} \) |
| 61 | \( 1 + 6.46T + 61T^{2} \) |
| 67 | \( 1 - 5.35T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 0.780T + 73T^{2} \) |
| 79 | \( 1 - 9.92T + 79T^{2} \) |
| 83 | \( 1 + 9.14T + 83T^{2} \) |
| 89 | \( 1 + 0.127T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
| 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
−7.85456610235851910689418869756, −6.88632674457972706262991545508, −5.95624391043293748479400771609, −5.55212548689666540027521929624, −5.13325720330534260723661893727, −4.04683619013105788953739303844, −3.22572112728790654069911731883, −2.30805169771804144271470409091, −1.71688801151493336907321421790, 0,
1.71688801151493336907321421790, 2.30805169771804144271470409091, 3.22572112728790654069911731883, 4.04683619013105788953739303844, 5.13325720330534260723661893727, 5.55212548689666540027521929624, 5.95624391043293748479400771609, 6.88632674457972706262991545508, 7.85456610235851910689418869756
