| L(s) = 1 | + 3-s − 1.86·7-s + 9-s + 0.675·11-s + 13-s + 5.31·17-s − 0.806·19-s − 1.86·21-s − 8.73·23-s + 27-s − 3.96·29-s − 1.71·31-s + 0.675·33-s − 7.38·37-s + 39-s − 8.54·41-s + 10.4·43-s + 2.13·47-s − 3.50·49-s + 5.31·51-s − 6.19·53-s − 0.806·57-s + 14.4·59-s − 10.3·61-s − 1.86·63-s + 1.06·67-s − 8.73·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.706·7-s + 0.333·9-s + 0.203·11-s + 0.277·13-s + 1.28·17-s − 0.184·19-s − 0.407·21-s − 1.82·23-s + 0.192·27-s − 0.735·29-s − 0.307·31-s + 0.117·33-s − 1.21·37-s + 0.160·39-s − 1.33·41-s + 1.59·43-s + 0.310·47-s − 0.500·49-s + 0.743·51-s − 0.850·53-s − 0.106·57-s + 1.88·59-s − 1.32·61-s − 0.235·63-s + 0.129·67-s − 1.05·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 1.86T + 7T^{2} \) |
| 11 | \( 1 - 0.675T + 11T^{2} \) |
| 17 | \( 1 - 5.31T + 17T^{2} \) |
| 19 | \( 1 + 0.806T + 19T^{2} \) |
| 23 | \( 1 + 8.73T + 23T^{2} \) |
| 29 | \( 1 + 3.96T + 29T^{2} \) |
| 31 | \( 1 + 1.71T + 31T^{2} \) |
| 37 | \( 1 + 7.38T + 37T^{2} \) |
| 41 | \( 1 + 8.54T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 2.13T + 47T^{2} \) |
| 53 | \( 1 + 6.19T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 1.06T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 8.34T + 73T^{2} \) |
| 79 | \( 1 + 6.93T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 - 0.261T + 97T^{2} \) |
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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
−7.57244203954822080946189165453, −6.91221513134643170505908712853, −6.04931469524107199889656781473, −5.60803903917434204237023926657, −4.53389517772733922673218354643, −3.66861186283513515364350152601, −3.31772121035556693787963817566, −2.24494302755815531601100852812, −1.40478154151444695496395476149, 0,
1.40478154151444695496395476149, 2.24494302755815531601100852812, 3.31772121035556693787963817566, 3.66861186283513515364350152601, 4.53389517772733922673218354643, 5.60803903917434204237023926657, 6.04931469524107199889656781473, 6.91221513134643170505908712853, 7.57244203954822080946189165453
