| L(s) = 1 | − 2-s − 4-s + 7-s + 3·8-s + 2·13-s − 14-s − 16-s − 2·17-s + 6·23-s − 2·26-s − 28-s + 10·29-s + 8·31-s − 5·32-s + 2·34-s − 4·37-s + 2·41-s − 8·43-s − 6·46-s + 6·47-s + 49-s − 2·52-s − 12·53-s + 3·56-s − 10·58-s − 4·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.377·7-s + 1.06·8-s + 0.554·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s + 1.25·23-s − 0.392·26-s − 0.188·28-s + 1.85·29-s + 1.43·31-s − 0.883·32-s + 0.342·34-s − 0.657·37-s + 0.312·41-s − 1.21·43-s − 0.884·46-s + 0.875·47-s + 1/7·49-s − 0.277·52-s − 1.64·53-s + 0.400·56-s − 1.31·58-s − 0.520·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.46916349980720, −12.86729865870159, −12.44182914052086, −11.88229163264684, −11.34085127632390, −10.85970786351501, −10.51279271665105, −9.986503937191835, −9.566954893077818, −8.967078469110531, −8.556315147585906, −8.296079133001077, −7.775326523683515, −7.161705483759957, −6.604600957547632, −6.278866109350259, −5.392900849188622, −4.938993650528121, −4.541631612910758, −4.057962072926074, −3.252595673315651, −2.792299821095883, −1.979922814811141, −1.230751258876272, −0.9045414923418024, 0,
0.9045414923418024, 1.230751258876272, 1.979922814811141, 2.792299821095883, 3.252595673315651, 4.057962072926074, 4.541631612910758, 4.938993650528121, 5.392900849188622, 6.278866109350259, 6.604600957547632, 7.161705483759957, 7.775326523683515, 8.296079133001077, 8.556315147585906, 8.967078469110531, 9.566954893077818, 9.986503937191835, 10.51279271665105, 10.85970786351501, 11.34085127632390, 11.88229163264684, 12.44182914052086, 12.86729865870159, 13.46916349980720
