L(s) = 1 | − 1.30·2-s − 0.302·4-s − 5-s + 7-s + 3·8-s + 1.30·10-s + 3.60·13-s − 1.30·14-s − 3.30·16-s + 4·17-s − 3·19-s + 0.302·20-s − 2·23-s − 4·25-s − 4.69·26-s − 0.302·28-s + 1.60·29-s − 2·31-s − 1.69·32-s − 5.21·34-s − 35-s + 6.21·37-s + 3.90·38-s − 3·40-s + 7.21·41-s − 9.21·43-s + 2.60·46-s + ⋯ |
L(s) = 1 | − 0.921·2-s − 0.151·4-s − 0.447·5-s + 0.377·7-s + 1.06·8-s + 0.411·10-s + 1.00·13-s − 0.348·14-s − 0.825·16-s + 0.970·17-s − 0.688·19-s + 0.0677·20-s − 0.417·23-s − 0.800·25-s − 0.921·26-s − 0.0572·28-s + 0.298·29-s − 0.359·31-s − 0.300·32-s − 0.893·34-s − 0.169·35-s + 1.02·37-s + 0.634·38-s − 0.474·40-s + 1.12·41-s − 1.40·43-s + 0.384·46-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 | \( 1 + 1.30T + 2T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 13 | \( 1 - 3.60T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 - 1.60T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 6.21T + 37T^{2} \) |
| 41 | \( 1 - 7.21T + 41T^{2} \) |
| 43 | \( 1 + 9.21T + 43T^{2} \) |
| 47 | \( 1 + 9.60T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 0.394T + 59T^{2} \) |
| 61 | \( 1 + 3.21T + 61T^{2} \) |
| 67 | \( 1 + 5.60T + 67T^{2} \) |
| 71 | \( 1 - 3.21T + 71T^{2} \) |
| 73 | \( 1 + 8.81T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−7.88519748319198779735712930753, −7.08582515219080393555793343909, −6.19470405567261666476807156427, −5.50011758096274979761931881595, −4.52914085245431927763304669668, −4.02346425942010706227572634743, −3.15440178102707882303401711315, −1.87950241539510524321478362508, −1.13535944883517024866479024893, 0,
1.13535944883517024866479024893, 1.87950241539510524321478362508, 3.15440178102707882303401711315, 4.02346425942010706227572634743, 4.52914085245431927763304669668, 5.50011758096274979761931881595, 6.19470405567261666476807156427, 7.08582515219080393555793343909, 7.88519748319198779735712930753