| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 7-s + 3·8-s − 2·9-s − 12-s − 2·13-s − 14-s − 16-s − 4·17-s + 2·18-s + 4·19-s + 21-s − 6·23-s + 3·24-s + 2·26-s − 5·27-s − 28-s − 9·29-s − 5·32-s + 4·34-s + 2·36-s + 3·37-s − 4·38-s − 2·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.377·7-s + 1.06·8-s − 2/3·9-s − 0.288·12-s − 0.554·13-s − 0.267·14-s − 1/4·16-s − 0.970·17-s + 0.471·18-s + 0.917·19-s + 0.218·21-s − 1.25·23-s + 0.612·24-s + 0.392·26-s − 0.962·27-s − 0.188·28-s − 1.67·29-s − 0.883·32-s + 0.685·34-s + 1/3·36-s + 0.493·37-s − 0.648·38-s − 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6854698737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6854698737\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−15.59075212848667, −14.92895856204898, −14.45806294699990, −14.03442288187537, −13.41449887143069, −13.13947965150187, −12.23572422536272, −11.67183331156875, −11.07865207787314, −10.60161862978036, −9.704058593121052, −9.464888864168756, −8.974945138512221, −8.203113481027384, −7.963451431917365, −7.377995287729666, −6.642540698177897, −5.642286923865513, −5.292006773635567, −4.364593557328010, −3.913920117031682, −3.029361722515952, −2.190226767820451, −1.584134284397204, −0.3630199993089607,
0.3630199993089607, 1.584134284397204, 2.190226767820451, 3.029361722515952, 3.913920117031682, 4.364593557328010, 5.292006773635567, 5.642286923865513, 6.642540698177897, 7.377995287729666, 7.963451431917365, 8.203113481027384, 8.974945138512221, 9.464888864168756, 9.704058593121052, 10.60161862978036, 11.07865207787314, 11.67183331156875, 12.23572422536272, 13.13947965150187, 13.41449887143069, 14.03442288187537, 14.45806294699990, 14.92895856204898, 15.59075212848667
