L(s) = 1 | − 2·2-s + 2·4-s − 7-s + 3·11-s + 13-s + 2·14-s − 4·16-s − 7·17-s − 6·22-s − 6·23-s − 2·26-s − 2·28-s + 5·29-s + 2·31-s + 8·32-s + 14·34-s + 2·37-s − 2·41-s − 4·43-s + 6·44-s + 12·46-s + 3·47-s + 49-s + 2·52-s − 6·53-s − 10·58-s − 10·59-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s − 0.377·7-s + 0.904·11-s + 0.277·13-s + 0.534·14-s − 16-s − 1.69·17-s − 1.27·22-s − 1.25·23-s − 0.392·26-s − 0.377·28-s + 0.928·29-s + 0.359·31-s + 1.41·32-s + 2.40·34-s + 0.328·37-s − 0.312·41-s − 0.609·43-s + 0.904·44-s + 1.76·46-s + 0.437·47-s + 1/7·49-s + 0.277·52-s − 0.824·53-s − 1.31·58-s − 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−9.047392015771699165995183825449, −8.428704975511724884581109770971, −7.67053504463822546742181996665, −6.59661422823789620490429923129, −6.33345798756843332553352393681, −4.74217082052326318941224012916, −3.90430059874868075785507099311, −2.48112200668525509199201292270, −1.41698761693361156521107028690, 0,
1.41698761693361156521107028690, 2.48112200668525509199201292270, 3.90430059874868075785507099311, 4.74217082052326318941224012916, 6.33345798756843332553352393681, 6.59661422823789620490429923129, 7.67053504463822546742181996665, 8.428704975511724884581109770971, 9.047392015771699165995183825449