L(s) = 1 | − 2·4-s − 5-s + 7-s + 3·11-s + 4·16-s − 3·17-s + 4·19-s + 2·20-s − 9·23-s + 25-s − 2·28-s − 6·29-s − 2·31-s − 35-s + 37-s + 3·41-s + 2·43-s − 6·44-s + 6·47-s − 6·49-s + 9·53-s − 3·55-s + 12·59-s + 5·61-s − 8·64-s + 4·67-s + 6·68-s + ⋯ |
L(s) = 1 | − 4-s − 0.447·5-s + 0.377·7-s + 0.904·11-s + 16-s − 0.727·17-s + 0.917·19-s + 0.447·20-s − 1.87·23-s + 1/5·25-s − 0.377·28-s − 1.11·29-s − 0.359·31-s − 0.169·35-s + 0.164·37-s + 0.468·41-s + 0.304·43-s − 0.904·44-s + 0.875·47-s − 6/7·49-s + 1.23·53-s − 0.404·55-s + 1.56·59-s + 0.640·61-s − 64-s + 0.488·67-s + 0.727·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−7.54495167213357747641176751824, −7.02972846457177953706217352267, −5.89571267239214679342187887171, −5.52033823119420140013004035083, −4.41746559741937451186206253760, −4.10389431963095048410347524916, −3.40228842974459432910683842492, −2.16030390386748734349732036514, −1.13579791971962286404590875048, 0,
1.13579791971962286404590875048, 2.16030390386748734349732036514, 3.40228842974459432910683842492, 4.10389431963095048410347524916, 4.41746559741937451186206253760, 5.52033823119420140013004035083, 5.89571267239214679342187887171, 7.02972846457177953706217352267, 7.54495167213357747641176751824