| L(s) = 1 | − 2.61·2-s − 3-s + 4.84·4-s − 5-s + 2.61·6-s + 3.80·7-s − 7.43·8-s + 9-s + 2.61·10-s − 2.35·11-s − 4.84·12-s + 1.16·13-s − 9.94·14-s + 15-s + 9.76·16-s − 7.43·17-s − 2.61·18-s − 7.79·19-s − 4.84·20-s − 3.80·21-s + 6.14·22-s − 2.57·23-s + 7.43·24-s + 25-s − 3.03·26-s − 27-s + 18.4·28-s + ⋯ |
| L(s) = 1 | − 1.84·2-s − 0.577·3-s + 2.42·4-s − 0.447·5-s + 1.06·6-s + 1.43·7-s − 2.62·8-s + 0.333·9-s + 0.827·10-s − 0.708·11-s − 1.39·12-s + 0.321·13-s − 2.65·14-s + 0.258·15-s + 2.44·16-s − 1.80·17-s − 0.616·18-s − 1.78·19-s − 1.08·20-s − 0.829·21-s + 1.31·22-s − 0.535·23-s + 1.51·24-s + 0.200·25-s − 0.595·26-s − 0.192·27-s + 3.47·28-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 | \( 1 + 2.61T + 2T^{2} \) |
| 7 | \( 1 - 3.80T + 7T^{2} \) |
| 11 | \( 1 + 2.35T + 11T^{2} \) |
| 13 | \( 1 - 1.16T + 13T^{2} \) |
| 17 | \( 1 + 7.43T + 17T^{2} \) |
| 19 | \( 1 + 7.79T + 19T^{2} \) |
| 23 | \( 1 + 2.57T + 23T^{2} \) |
| 29 | \( 1 - 7.64T + 29T^{2} \) |
| 31 | \( 1 - 9.13T + 31T^{2} \) |
| 37 | \( 1 - 7.38T + 37T^{2} \) |
| 41 | \( 1 + 0.798T + 41T^{2} \) |
| 43 | \( 1 + 6.13T + 43T^{2} \) |
| 47 | \( 1 - 4.48T + 47T^{2} \) |
| 53 | \( 1 - 0.880T + 53T^{2} \) |
| 59 | \( 1 - 4.50T + 59T^{2} \) |
| 61 | \( 1 - 4.55T + 61T^{2} \) |
| 67 | \( 1 - 1.67T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 + 9.92T + 83T^{2} \) |
| 89 | \( 1 - 3.61T + 89T^{2} \) |
| 97 | \( 1 + 19.4T + 97T^{2} \) |
| 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
−8.102390641333197726052749088062, −7.17502061773644243296856651603, −6.57788835328464590885474669106, −5.97214137987708718782745452304, −4.70293008095732860268021226137, −4.31027824483255626837092143564, −2.59029470865520953218737861436, −2.05579453624886438674847549358, −1.01429625832514483278583805277, 0,
1.01429625832514483278583805277, 2.05579453624886438674847549358, 2.59029470865520953218737861436, 4.31027824483255626837092143564, 4.70293008095732860268021226137, 5.97214137987708718782745452304, 6.57788835328464590885474669106, 7.17502061773644243296856651603, 8.102390641333197726052749088062
