| L(s) = 1 | + 1.75·2-s + 3-s + 1.09·4-s + 5-s + 1.75·6-s − 3.35·7-s − 1.59·8-s + 9-s + 1.75·10-s + 4.95·11-s + 1.09·12-s − 4.38·13-s − 5.90·14-s + 15-s − 4.99·16-s + 0.849·17-s + 1.75·18-s − 6.37·19-s + 1.09·20-s − 3.35·21-s + 8.71·22-s + 2.99·23-s − 1.59·24-s + 25-s − 7.71·26-s + 27-s − 3.67·28-s + ⋯ |
| L(s) = 1 | + 1.24·2-s + 0.577·3-s + 0.548·4-s + 0.447·5-s + 0.718·6-s − 1.26·7-s − 0.562·8-s + 0.333·9-s + 0.556·10-s + 1.49·11-s + 0.316·12-s − 1.21·13-s − 1.57·14-s + 0.258·15-s − 1.24·16-s + 0.206·17-s + 0.414·18-s − 1.46·19-s + 0.245·20-s − 0.732·21-s + 1.85·22-s + 0.624·23-s − 0.324·24-s + 0.200·25-s − 1.51·26-s + 0.192·27-s − 0.695·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 1.75T + 2T^{2} \) |
| 7 | \( 1 + 3.35T + 7T^{2} \) |
| 11 | \( 1 - 4.95T + 11T^{2} \) |
| 13 | \( 1 + 4.38T + 13T^{2} \) |
| 17 | \( 1 - 0.849T + 17T^{2} \) |
| 19 | \( 1 + 6.37T + 19T^{2} \) |
| 23 | \( 1 - 2.99T + 23T^{2} \) |
| 29 | \( 1 + 1.30T + 29T^{2} \) |
| 31 | \( 1 + 4.99T + 31T^{2} \) |
| 37 | \( 1 + 2.12T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 - 3.14T + 43T^{2} \) |
| 47 | \( 1 - 7.47T + 47T^{2} \) |
| 53 | \( 1 + 7.02T + 53T^{2} \) |
| 59 | \( 1 + 4.59T + 59T^{2} \) |
| 61 | \( 1 + 3.05T + 61T^{2} \) |
| 67 | \( 1 - 3.08T + 67T^{2} \) |
| 71 | \( 1 + 5.13T + 71T^{2} \) |
| 73 | \( 1 + 7.08T + 73T^{2} \) |
| 79 | \( 1 + 6.02T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 0.595T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−7.38101281233254800666043590641, −6.74877790752705264870670562158, −6.31795132510483438642773307386, −5.56094769769137146207755753984, −4.65489821125903751802526280051, −4.01444420621522701413105382612, −3.33444914186832811296927151059, −2.67766787023837012215842701360, −1.74062391523159567069542106426, 0,
1.74062391523159567069542106426, 2.67766787023837012215842701360, 3.33444914186832811296927151059, 4.01444420621522701413105382612, 4.65489821125903751802526280051, 5.56094769769137146207755753984, 6.31795132510483438642773307386, 6.74877790752705264870670562158, 7.38101281233254800666043590641
