L(s) = 1 | − 3-s + 5-s − 7-s + 9-s + 4·11-s − 2·13-s − 15-s − 6·17-s + 21-s − 23-s + 25-s − 27-s − 2·29-s − 8·31-s − 4·33-s − 35-s + 2·37-s + 2·39-s + 2·41-s + 4·43-s + 45-s + 8·47-s + 49-s + 6·51-s + 6·53-s + 4·55-s − 8·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s − 1.45·17-s + 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.696·33-s − 0.169·35-s + 0.328·37-s + 0.320·39-s + 0.312·41-s + 0.609·43-s + 0.149·45-s + 1.16·47-s + 1/7·49-s + 0.840·51-s + 0.824·53-s + 0.539·55-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.97945025734039, −14.68593512516556, −13.95612403041576, −13.53380775811027, −12.96576823873608, −12.44078821491605, −11.99341552302259, −11.43414023061536, −10.77367791463153, −10.57824386554079, −9.652594449220384, −9.177822011082514, −9.069085585674030, −8.120085426094459, −7.287819668269074, −6.969209386438080, −6.328449493789736, −5.888641948134181, −5.284253214702878, −4.462720465127969, −4.090165738091516, −3.333846753458655, −2.359374358104211, −1.872737870804002, −0.9273596356821832, 0,
0.9273596356821832, 1.872737870804002, 2.359374358104211, 3.333846753458655, 4.090165738091516, 4.462720465127969, 5.284253214702878, 5.888641948134181, 6.328449493789736, 6.969209386438080, 7.287819668269074, 8.120085426094459, 9.069085585674030, 9.177822011082514, 9.652594449220384, 10.57824386554079, 10.77367791463153, 11.43414023061536, 11.99341552302259, 12.44078821491605, 12.96576823873608, 13.53380775811027, 13.95612403041576, 14.68593512516556, 14.97945025734039