| L(s) = 1 | − 2-s + 3-s + 4-s − 3·5-s − 6-s − 7-s − 8-s + 9-s + 3·10-s − 3·11-s + 12-s − 13-s + 14-s − 3·15-s + 16-s − 6·17-s − 18-s − 4·19-s − 3·20-s − 21-s + 3·22-s + 3·23-s − 24-s + 4·25-s + 26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.904·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.670·20-s − 0.218·21-s + 0.639·22-s + 0.625·23-s − 0.204·24-s + 4/5·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 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
−10.93257693723723125654778976942, −10.00355305179054067338945705437, −8.872827440745571819543828173395, −8.251185706791234441455516215602, −7.40299316320484687999199352794, −6.56604464795087490836127725412, −4.79576132046549101101034444564, −3.61477936593783888825584024570, −2.38302524708346316952707169278, 0,
2.38302524708346316952707169278, 3.61477936593783888825584024570, 4.79576132046549101101034444564, 6.56604464795087490836127725412, 7.40299316320484687999199352794, 8.251185706791234441455516215602, 8.872827440745571819543828173395, 10.00355305179054067338945705437, 10.93257693723723125654778976942
