| L(s) = 1 | − 2.30·2-s + 3.30·4-s − 3.60·5-s − 7-s − 3.00·8-s + 8.30·10-s + 6.60·13-s + 2.30·14-s + 0.302·16-s − 2.69·17-s − 3·19-s − 11.9·20-s + 2.69·23-s + 7.99·25-s − 15.2·26-s − 3.30·28-s − 4.69·29-s + 31-s + 5.30·32-s + 6.21·34-s + 3.60·35-s − 5.21·37-s + 6.90·38-s + 10.8·40-s − 7·41-s + 1.69·43-s − 6.21·46-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 1.65·4-s − 1.61·5-s − 0.377·7-s − 1.06·8-s + 2.62·10-s + 1.83·13-s + 0.615·14-s + 0.0756·16-s − 0.654·17-s − 0.688·19-s − 2.66·20-s + 0.562·23-s + 1.59·25-s − 2.98·26-s − 0.624·28-s − 0.872·29-s + 0.179·31-s + 0.937·32-s + 1.06·34-s + 0.609·35-s − 0.856·37-s + 1.12·38-s + 1.71·40-s − 1.09·41-s + 0.258·43-s − 0.915·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 5 | \( 1 + 3.60T + 5T^{2} \) |
| 13 | \( 1 - 6.60T + 13T^{2} \) |
| 17 | \( 1 + 2.69T + 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 - 2.69T + 23T^{2} \) |
| 29 | \( 1 + 4.69T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + 5.21T + 37T^{2} \) |
| 41 | \( 1 + 7T + 41T^{2} \) |
| 43 | \( 1 - 1.69T + 43T^{2} \) |
| 47 | \( 1 - 1.90T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 6.69T + 59T^{2} \) |
| 61 | \( 1 + 4.30T + 61T^{2} \) |
| 67 | \( 1 - 8.51T + 67T^{2} \) |
| 71 | \( 1 - 4.30T + 71T^{2} \) |
| 73 | \( 1 + 5T + 73T^{2} \) |
| 79 | \( 1 + 8.30T + 79T^{2} \) |
| 83 | \( 1 + 3T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 3.60T + 97T^{2} \) |
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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
−7.77758870980026191443477923205, −6.95244404217706450358545438050, −6.67213960577919432482312127610, −5.67547434115396659549410619232, −4.43732231556913125379368389792, −3.80673019137258722371733740744, −3.08758312299516695693876691755, −1.89080315756901568626900252622, −0.880751857162618830403849390093, 0,
0.880751857162618830403849390093, 1.89080315756901568626900252622, 3.08758312299516695693876691755, 3.80673019137258722371733740744, 4.43732231556913125379368389792, 5.67547434115396659549410619232, 6.67213960577919432482312127610, 6.95244404217706450358545438050, 7.77758870980026191443477923205
