| L(s) = 1 | + 3-s − 7-s − 2·9-s + 2·11-s − 5·13-s − 17-s − 19-s − 21-s − 5·27-s + 29-s − 7·31-s + 2·33-s + 8·37-s − 5·39-s + 12·41-s + 4·43-s + 47-s + 49-s − 51-s − 3·53-s − 57-s − 5·59-s + 11·61-s + 2·63-s − 2·67-s + 71-s − 73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.603·11-s − 1.38·13-s − 0.242·17-s − 0.229·19-s − 0.218·21-s − 0.962·27-s + 0.185·29-s − 1.25·31-s + 0.348·33-s + 1.31·37-s − 0.800·39-s + 1.87·41-s + 0.609·43-s + 0.145·47-s + 1/7·49-s − 0.140·51-s − 0.412·53-s − 0.132·57-s − 0.650·59-s + 1.40·61-s + 0.251·63-s − 0.244·67-s + 0.118·71-s − 0.117·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−14.67349816619838, −14.40262067584368, −14.02106794805357, −13.23741055874891, −12.80506722294829, −12.36236340043915, −11.75912403804713, −11.20704415054956, −10.78746069648808, −9.991000393219623, −9.450718119505502, −9.201469014794186, −8.641194993843008, −7.879012116114235, −7.520468363229340, −6.945125689670303, −6.200153262864930, −5.779720147288883, −5.058798680350150, −4.354982227388774, −3.832188036003275, −3.080592276436550, −2.481522534229385, −2.052963818154827, −0.8978633196616060, 0,
0.8978633196616060, 2.052963818154827, 2.481522534229385, 3.080592276436550, 3.832188036003275, 4.354982227388774, 5.058798680350150, 5.779720147288883, 6.200153262864930, 6.945125689670303, 7.520468363229340, 7.879012116114235, 8.641194993843008, 9.201469014794186, 9.450718119505502, 9.991000393219623, 10.78746069648808, 11.20704415054956, 11.75912403804713, 12.36236340043915, 12.80506722294829, 13.23741055874891, 14.02106794805357, 14.40262067584368, 14.67349816619838
