| L(s) = 1 | + 3.48·3-s − 59.8·5-s − 145.·7-s − 230.·9-s − 121·11-s + 615.·13-s − 208.·15-s + 1.84e3·17-s − 366.·19-s − 505.·21-s + 4.51e3·23-s + 459.·25-s − 1.65e3·27-s − 1.71e3·29-s + 2.65e3·31-s − 421.·33-s + 8.68e3·35-s + 9.66e3·37-s + 2.14e3·39-s − 1.11e4·41-s − 8.36e3·43-s + 1.38e4·45-s + 2.22e3·47-s + 4.23e3·49-s + 6.41e3·51-s + 2.37e4·53-s + 7.24e3·55-s + ⋯ |
| L(s) = 1 | + 0.223·3-s − 1.07·5-s − 1.11·7-s − 0.949·9-s − 0.301·11-s + 1.01·13-s − 0.239·15-s + 1.54·17-s − 0.232·19-s − 0.250·21-s + 1.78·23-s + 0.147·25-s − 0.436·27-s − 0.379·29-s + 0.495·31-s − 0.0674·33-s + 1.19·35-s + 1.16·37-s + 0.225·39-s − 1.03·41-s − 0.690·43-s + 1.01·45-s + 0.146·47-s + 0.252·49-s + 0.345·51-s + 1.15·53-s + 0.322·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.223636638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.223636638\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 3 | \( 1 - 3.48T + 243T^{2} \) |
| 5 | \( 1 + 59.8T + 3.12e3T^{2} \) |
| 7 | \( 1 + 145.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 615.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.84e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 366.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.51e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.71e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.65e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.66e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.11e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.36e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.22e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.37e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.95e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.09e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.17e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.39e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.24e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.46e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.67e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.76e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.83e4T + 8.58e9T^{2} \) |
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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
−11.77237765668583924635433281482, −10.95772089600745345903165405349, −9.744956639385253730499668237130, −8.642790166111257396934610847869, −7.81582666584048064888033322354, −6.59162039441687536518210681964, −5.40429490645093890308204859268, −3.68868556020357652724600750105, −3.00324836445701566840171319838, −0.68115080275392638450641550873,
0.68115080275392638450641550873, 3.00324836445701566840171319838, 3.68868556020357652724600750105, 5.40429490645093890308204859268, 6.59162039441687536518210681964, 7.81582666584048064888033322354, 8.642790166111257396934610847869, 9.744956639385253730499668237130, 10.95772089600745345903165405349, 11.77237765668583924635433281482
