| L(s) = 1 | − 2-s − 3-s − 4-s + 6-s + 3·8-s + 9-s + 11-s + 12-s − 2·13-s − 16-s − 2·17-s − 18-s − 22-s − 8·23-s − 3·24-s + 2·26-s − 27-s − 6·29-s + 8·31-s − 5·32-s − 33-s + 2·34-s − 36-s − 6·37-s + 2·39-s + 2·41-s − 44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s − 0.554·13-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.213·22-s − 1.66·23-s − 0.612·24-s + 0.392·26-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.883·32-s − 0.174·33-s + 0.342·34-s − 1/6·36-s − 0.986·37-s + 0.320·39-s + 0.312·41-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.06839898370076, −14.46063913636934, −13.91944057038330, −13.54946466409561, −12.92669765683386, −12.31588986725397, −11.92105720908832, −11.32496002311413, −10.70134183724355, −10.19194322579723, −9.836340510925709, −9.232282622814423, −8.736492542686742, −8.127087016923967, −7.563814822849048, −7.122556843295325, −6.324674917374737, −5.840086653096625, −5.135179027316095, −4.493608447860747, −4.107679977271536, −3.331815423518953, −2.233990191375352, −1.683190300847068, −0.7262044861124459, 0,
0.7262044861124459, 1.683190300847068, 2.233990191375352, 3.331815423518953, 4.107679977271536, 4.493608447860747, 5.135179027316095, 5.840086653096625, 6.324674917374737, 7.122556843295325, 7.563814822849048, 8.127087016923967, 8.736492542686742, 9.232282622814423, 9.836340510925709, 10.19194322579723, 10.70134183724355, 11.32496002311413, 11.92105720908832, 12.31588986725397, 12.92669765683386, 13.54946466409561, 13.91944057038330, 14.46063913636934, 15.06839898370076
