L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 5-s − 2·6-s − 7-s + 9-s − 2·10-s − 11-s − 2·12-s − 2·13-s − 2·14-s + 15-s − 4·16-s − 3·17-s + 2·18-s − 5·19-s − 2·20-s + 21-s − 2·22-s + 3·23-s + 25-s − 4·26-s − 27-s − 2·28-s − 3·29-s + 2·30-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s − 0.377·7-s + 1/3·9-s − 0.632·10-s − 0.301·11-s − 0.577·12-s − 0.554·13-s − 0.534·14-s + 0.258·15-s − 16-s − 0.727·17-s + 0.471·18-s − 1.14·19-s − 0.447·20-s + 0.218·21-s − 0.426·22-s + 0.625·23-s + 1/5·25-s − 0.784·26-s − 0.192·27-s − 0.377·28-s − 0.557·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−9.422001093043397186804010524168, −8.563023038477538009830995801312, −7.30863462583217475084211140545, −6.64804517128032150253156126362, −5.83924706607087041773981749973, −4.94680144297311481675050269108, −4.30376854047693015838887036907, −3.35922219277480043497832113715, −2.24141850867322818905034178851, 0,
2.24141850867322818905034178851, 3.35922219277480043497832113715, 4.30376854047693015838887036907, 4.94680144297311481675050269108, 5.83924706607087041773981749973, 6.64804517128032150253156126362, 7.30863462583217475084211140545, 8.563023038477538009830995801312, 9.422001093043397186804010524168