L(s) = 1 | + 4·2-s + 2·3-s + 8·4-s − 5·5-s + 8·6-s − 6·7-s − 23·9-s − 20·10-s + 16·12-s + 38·13-s − 24·14-s − 10·15-s − 64·16-s − 26·17-s − 92·18-s − 100·19-s − 40·20-s − 12·21-s − 78·23-s + 25·25-s + 152·26-s − 100·27-s − 48·28-s + 50·29-s − 40·30-s − 108·31-s − 256·32-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 0.384·3-s + 4-s − 0.447·5-s + 0.544·6-s − 0.323·7-s − 0.851·9-s − 0.632·10-s + 0.384·12-s + 0.810·13-s − 0.458·14-s − 0.172·15-s − 16-s − 0.370·17-s − 1.20·18-s − 1.20·19-s − 0.447·20-s − 0.124·21-s − 0.707·23-s + 1/5·25-s + 1.14·26-s − 0.712·27-s − 0.323·28-s + 0.320·29-s − 0.243·30-s − 0.625·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + p T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 7 | \( 1 + 6 T + p^{3} T^{2} \) |
| 13 | \( 1 - 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 26 T + p^{3} T^{2} \) |
| 19 | \( 1 + 100 T + p^{3} T^{2} \) |
| 23 | \( 1 + 78 T + p^{3} T^{2} \) |
| 29 | \( 1 - 50 T + p^{3} T^{2} \) |
| 31 | \( 1 + 108 T + p^{3} T^{2} \) |
| 37 | \( 1 - 266 T + p^{3} T^{2} \) |
| 41 | \( 1 + 22 T + p^{3} T^{2} \) |
| 43 | \( 1 + 442 T + p^{3} T^{2} \) |
| 47 | \( 1 + 514 T + p^{3} T^{2} \) |
| 53 | \( 1 - 2 T + p^{3} T^{2} \) |
| 59 | \( 1 - 500 T + p^{3} T^{2} \) |
| 61 | \( 1 - 518 T + p^{3} T^{2} \) |
| 67 | \( 1 - 126 T + p^{3} T^{2} \) |
| 71 | \( 1 - 412 T + p^{3} T^{2} \) |
| 73 | \( 1 - 878 T + p^{3} T^{2} \) |
| 79 | \( 1 + 600 T + p^{3} T^{2} \) |
| 83 | \( 1 + 282 T + p^{3} T^{2} \) |
| 89 | \( 1 + 150 T + p^{3} T^{2} \) |
| 97 | \( 1 - 386 T + p^{3} T^{2} \) |
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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
−9.859136641615114443745606461166, −8.712171887583194838916139914552, −8.153990620441612553838563025613, −6.68607495135358755463518225545, −6.10240764845729438287117548498, −5.03793274786996369406823320201, −3.98524887702113779285395319249, −3.30783837142907774635534987041, −2.20888740482190146568794437978, 0,
2.20888740482190146568794437978, 3.30783837142907774635534987041, 3.98524887702113779285395319249, 5.03793274786996369406823320201, 6.10240764845729438287117548498, 6.68607495135358755463518225545, 8.153990620441612553838563025613, 8.712171887583194838916139914552, 9.859136641615114443745606461166