| L(s) = 1 | + 3-s − 5·5-s − 7·7-s − 26·9-s − 7·11-s − 23·13-s − 5·15-s − 25·17-s − 62·19-s − 7·21-s − 86·23-s + 25·25-s − 53·27-s − 29·29-s − 12·31-s − 7·33-s + 35·35-s − 150·37-s − 23·39-s + 204·41-s − 178·43-s + 130·45-s + 33·47-s + 49·49-s − 25·51-s + 452·53-s + 35·55-s + ⋯ |
| L(s) = 1 | + 0.192·3-s − 0.447·5-s − 0.377·7-s − 0.962·9-s − 0.191·11-s − 0.490·13-s − 0.0860·15-s − 0.356·17-s − 0.748·19-s − 0.0727·21-s − 0.779·23-s + 1/5·25-s − 0.377·27-s − 0.185·29-s − 0.0695·31-s − 0.0369·33-s + 0.169·35-s − 0.666·37-s − 0.0944·39-s + 0.777·41-s − 0.631·43-s + 0.430·45-s + 0.102·47-s + 1/7·49-s − 0.0686·51-s + 1.17·53-s + 0.0858·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( 1 - T + p^{3} T^{2} \) |
| 11 | \( 1 + 7 T + p^{3} T^{2} \) |
| 13 | \( 1 + 23 T + p^{3} T^{2} \) |
| 17 | \( 1 + 25 T + p^{3} T^{2} \) |
| 19 | \( 1 + 62 T + p^{3} T^{2} \) |
| 23 | \( 1 + 86 T + p^{3} T^{2} \) |
| 29 | \( 1 + p T + p^{3} T^{2} \) |
| 31 | \( 1 + 12 T + p^{3} T^{2} \) |
| 37 | \( 1 + 150 T + p^{3} T^{2} \) |
| 41 | \( 1 - 204 T + p^{3} T^{2} \) |
| 43 | \( 1 + 178 T + p^{3} T^{2} \) |
| 47 | \( 1 - 33 T + p^{3} T^{2} \) |
| 53 | \( 1 - 452 T + p^{3} T^{2} \) |
| 59 | \( 1 - 120 T + p^{3} T^{2} \) |
| 61 | \( 1 - 920 T + p^{3} T^{2} \) |
| 67 | \( 1 + 300 T + p^{3} T^{2} \) |
| 71 | \( 1 - 520 T + p^{3} T^{2} \) |
| 73 | \( 1 - 370 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1013 T + p^{3} T^{2} \) |
| 83 | \( 1 + 636 T + p^{3} T^{2} \) |
| 89 | \( 1 - 292 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1381 T + p^{3} T^{2} \) |
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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
−12.15392766042173395873366157015, −11.23485670719268420322659886864, −10.13166433655185268065485675914, −8.917324212997053450282997915212, −8.025156938182668703859375306741, −6.75768427767400717106404295277, −5.46867210456653611258595840043, −3.94167995369757537700192233173, −2.50502383480248401294588602370, 0,
2.50502383480248401294588602370, 3.94167995369757537700192233173, 5.46867210456653611258595840043, 6.75768427767400717106404295277, 8.025156938182668703859375306741, 8.917324212997053450282997915212, 10.13166433655185268065485675914, 11.23485670719268420322659886864, 12.15392766042173395873366157015
