| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 3·7-s − 8-s − 2·9-s + 10-s − 4·11-s + 12-s + 13-s − 3·14-s − 15-s + 16-s + 3·17-s + 2·18-s − 19-s − 20-s + 3·21-s + 4·22-s + 6·23-s − 24-s + 25-s − 26-s − 5·27-s + 3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.13·7-s − 0.353·8-s − 2/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.277·13-s − 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.471·18-s − 0.229·19-s − 0.223·20-s + 0.654·21-s + 0.852·22-s + 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.962·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 139 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−14.95486317339929, −14.67625107436213, −14.14269456945371, −13.56135982068555, −12.91080623256468, −12.41423262689352, −11.76072488967738, −11.20493781009010, −10.77172734489508, −10.56604019574362, −9.517795771652356, −9.179414053932232, −8.549494165314268, −8.060580727219621, −7.725950956092481, −7.371234662236985, −6.433903462462327, −5.753276611394353, −5.125109121936530, −4.678834635076168, −3.685308434405566, −3.077654345363257, −2.532371233804428, −1.785982111638474, −0.9808344236120019, 0,
0.9808344236120019, 1.785982111638474, 2.532371233804428, 3.077654345363257, 3.685308434405566, 4.678834635076168, 5.125109121936530, 5.753276611394353, 6.433903462462327, 7.371234662236985, 7.725950956092481, 8.060580727219621, 8.549494165314268, 9.179414053932232, 9.517795771652356, 10.56604019574362, 10.77172734489508, 11.20493781009010, 11.76072488967738, 12.41423262689352, 12.91080623256468, 13.56135982068555, 14.14269456945371, 14.67625107436213, 14.95486317339929
