| L(s) = 1 | − 2-s − 3-s + 4-s − 4·5-s + 6-s − 2·7-s − 8-s + 9-s + 4·10-s − 12-s − 6·13-s + 2·14-s + 4·15-s + 16-s − 17-s − 18-s + 4·19-s − 4·20-s + 2·21-s + 6·23-s + 24-s + 11·25-s + 6·26-s − 27-s − 2·28-s − 4·29-s − 4·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 1.26·10-s − 0.288·12-s − 1.66·13-s + 0.534·14-s + 1.03·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.894·20-s + 0.436·21-s + 1.25·23-s + 0.204·24-s + 11/5·25-s + 1.17·26-s − 0.192·27-s − 0.377·28-s − 0.742·29-s − 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.85996017417380940462743344925, −12.00111293806257788946097141198, −11.31840597033055557513179299020, −10.13636675520507553497406453887, −8.951168622566808298087877447389, −7.50224155158573118874050462938, −6.99640780173569532110131873676, −5.00716852164944696644803170219, −3.34589616522309311397390467846, 0,
3.34589616522309311397390467846, 5.00716852164944696644803170219, 6.99640780173569532110131873676, 7.50224155158573118874050462938, 8.951168622566808298087877447389, 10.13636675520507553497406453887, 11.31840597033055557513179299020, 12.00111293806257788946097141198, 12.85996017417380940462743344925
