L(s) = 1 | − 2-s + 2.32·3-s + 4-s + 1.22·5-s − 2.32·6-s − 2.49·7-s − 8-s + 2.40·9-s − 1.22·10-s − 6.12·11-s + 2.32·12-s − 4.23·13-s + 2.49·14-s + 2.84·15-s + 16-s − 3.48·17-s − 2.40·18-s − 0.335·19-s + 1.22·20-s − 5.80·21-s + 6.12·22-s + 23-s − 2.32·24-s − 3.50·25-s + 4.23·26-s − 1.37·27-s − 2.49·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.34·3-s + 0.5·4-s + 0.547·5-s − 0.949·6-s − 0.942·7-s − 0.353·8-s + 0.802·9-s − 0.387·10-s − 1.84·11-s + 0.671·12-s − 1.17·13-s + 0.666·14-s + 0.735·15-s + 0.250·16-s − 0.844·17-s − 0.567·18-s − 0.0770·19-s + 0.273·20-s − 1.26·21-s + 1.30·22-s + 0.208·23-s − 0.474·24-s − 0.700·25-s + 0.831·26-s − 0.264·27-s − 0.471·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
good | 3 | \( 1 - 2.32T + 3T^{2} \) |
| 5 | \( 1 - 1.22T + 5T^{2} \) |
| 7 | \( 1 + 2.49T + 7T^{2} \) |
| 11 | \( 1 + 6.12T + 11T^{2} \) |
| 13 | \( 1 + 4.23T + 13T^{2} \) |
| 17 | \( 1 + 3.48T + 17T^{2} \) |
| 19 | \( 1 + 0.335T + 19T^{2} \) |
| 31 | \( 1 + 6.86T + 31T^{2} \) |
| 37 | \( 1 - 4.36T + 37T^{2} \) |
| 41 | \( 1 + 1.68T + 41T^{2} \) |
| 43 | \( 1 - 0.798T + 43T^{2} \) |
| 47 | \( 1 - 9.07T + 47T^{2} \) |
| 53 | \( 1 + 7.30T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 - 7.57T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 + 16.0T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 + 6.85T + 97T^{2} \) |
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
−9.242578524563633656438575589193, −8.522073847386230796662185835454, −7.66717967220935484685618151569, −7.17774567684440791496436179983, −6.02073922220545041077559287206, −5.04356918860612027975406636387, −3.61049994991494313934052127874, −2.50285312409096883899484087150, −2.29770890664454914422425095925, 0,
2.29770890664454914422425095925, 2.50285312409096883899484087150, 3.61049994991494313934052127874, 5.04356918860612027975406636387, 6.02073922220545041077559287206, 7.17774567684440791496436179983, 7.66717967220935484685618151569, 8.522073847386230796662185835454, 9.242578524563633656438575589193