L(s) = 1 | + 1.49·2-s − 3-s + 0.223·4-s − 1.49·6-s − 1.33·7-s − 2.64·8-s + 9-s + 3.95·11-s − 0.223·12-s − 0.0410·13-s − 1.99·14-s − 4.39·16-s + 6.69·17-s + 1.49·18-s − 5.69·19-s + 1.33·21-s + 5.89·22-s − 4.16·23-s + 2.64·24-s − 0.0612·26-s − 27-s − 0.299·28-s − 7.48·29-s + 0.727·31-s − 1.26·32-s − 3.95·33-s + 9.98·34-s + ⋯ |
L(s) = 1 | + 1.05·2-s − 0.577·3-s + 0.111·4-s − 0.608·6-s − 0.504·7-s − 0.936·8-s + 0.333·9-s + 1.19·11-s − 0.0646·12-s − 0.0113·13-s − 0.532·14-s − 1.09·16-s + 1.62·17-s + 0.351·18-s − 1.30·19-s + 0.291·21-s + 1.25·22-s − 0.868·23-s + 0.540·24-s − 0.0120·26-s − 0.192·27-s − 0.0565·28-s − 1.38·29-s + 0.130·31-s − 0.222·32-s − 0.688·33-s + 1.71·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 47 | \( 1 - T \) |
good | 2 | \( 1 - 1.49T + 2T^{2} \) |
| 7 | \( 1 + 1.33T + 7T^{2} \) |
| 11 | \( 1 - 3.95T + 11T^{2} \) |
| 13 | \( 1 + 0.0410T + 13T^{2} \) |
| 17 | \( 1 - 6.69T + 17T^{2} \) |
| 19 | \( 1 + 5.69T + 19T^{2} \) |
| 23 | \( 1 + 4.16T + 23T^{2} \) |
| 29 | \( 1 + 7.48T + 29T^{2} \) |
| 31 | \( 1 - 0.727T + 31T^{2} \) |
| 37 | \( 1 - 5.03T + 37T^{2} \) |
| 41 | \( 1 + 2.90T + 41T^{2} \) |
| 43 | \( 1 - 5.70T + 43T^{2} \) |
| 53 | \( 1 - 5.79T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 + 4.38T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 4.68T + 89T^{2} \) |
| 97 | \( 1 + 8.64T + 97T^{2} \) |
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
−8.115041022234262412620009725656, −7.23349977513363709599238864229, −6.17080029289565358098226601506, −6.09963996934055767288207703369, −5.18779358858130387446133674204, −4.18739579583193658619157738391, −3.83098995659768309847622130725, −2.84971835674577985769433866448, −1.49096638385107611570910252512, 0,
1.49096638385107611570910252512, 2.84971835674577985769433866448, 3.83098995659768309847622130725, 4.18739579583193658619157738391, 5.18779358858130387446133674204, 6.09963996934055767288207703369, 6.17080029289565358098226601506, 7.23349977513363709599238864229, 8.115041022234262412620009725656