| L(s) = 1 | + 5-s − 2·7-s − 2·11-s − 2·17-s − 4·19-s + 25-s + 2·29-s − 8·31-s − 2·35-s − 4·37-s − 8·41-s − 8·43-s + 8·47-s − 3·49-s + 10·53-s − 2·55-s + 6·59-s + 2·61-s − 12·67-s − 12·71-s − 2·73-s + 4·77-s − 8·79-s + 4·83-s − 2·85-s − 12·89-s − 4·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 0.603·11-s − 0.485·17-s − 0.917·19-s + 1/5·25-s + 0.371·29-s − 1.43·31-s − 0.338·35-s − 0.657·37-s − 1.24·41-s − 1.21·43-s + 1.16·47-s − 3/7·49-s + 1.37·53-s − 0.269·55-s + 0.781·59-s + 0.256·61-s − 1.46·67-s − 1.42·71-s − 0.234·73-s + 0.455·77-s − 0.900·79-s + 0.439·83-s − 0.216·85-s − 1.27·89-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.008791119141243942717159667737, −8.564085609686467345322463540902, −7.37242345424133052115191128471, −6.68289289853968084678108249114, −5.85349627514271365714593881393, −5.01909859274436078455320866302, −3.91480640522384398014282785525, −2.88838332187413604660369820241, −1.84983133398470330100799616342, 0,
1.84983133398470330100799616342, 2.88838332187413604660369820241, 3.91480640522384398014282785525, 5.01909859274436078455320866302, 5.85349627514271365714593881393, 6.68289289853968084678108249114, 7.37242345424133052115191128471, 8.564085609686467345322463540902, 9.008791119141243942717159667737
