| L(s) = 1 | + (−1.30 − 0.545i)2-s + (0.319 − 2.22i)3-s + (1.40 + 1.42i)4-s + (−0.959 + 0.281i)5-s + (−1.63 + 2.72i)6-s + (0.470 + 0.542i)7-s + (−1.05 − 2.62i)8-s + (−1.96 − 0.577i)9-s + (1.40 + 0.155i)10-s + (0.0968 − 0.150i)11-s + (3.61 − 2.67i)12-s + (4.77 + 4.13i)13-s + (−0.317 − 0.964i)14-s + (0.319 + 2.22i)15-s + (−0.0517 + 3.99i)16-s + (0.839 − 0.383i)17-s + ⋯ |
| L(s) = 1 | + (−0.922 − 0.385i)2-s + (0.184 − 1.28i)3-s + (0.702 + 0.711i)4-s + (−0.429 + 0.125i)5-s + (−0.665 + 1.11i)6-s + (0.177 + 0.205i)7-s + (−0.373 − 0.927i)8-s + (−0.655 − 0.192i)9-s + (0.444 + 0.0492i)10-s + (0.0292 − 0.0454i)11-s + (1.04 − 0.770i)12-s + (1.32 + 1.14i)13-s + (−0.0848 − 0.257i)14-s + (0.0825 + 0.574i)15-s + (−0.0129 + 0.999i)16-s + (0.203 − 0.0930i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.883197 - 0.706812i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.883197 - 0.706812i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.30 + 0.545i)T \) |
| 5 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (-4.51 + 1.61i)T \) |
| good | 3 | \( 1 + (-0.319 + 2.22i)T + (-2.87 - 0.845i)T^{2} \) |
| 7 | \( 1 + (-0.470 - 0.542i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-0.0968 + 0.150i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-4.77 - 4.13i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.839 + 0.383i)T + (11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (0.153 + 0.0699i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-4.18 + 1.91i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (5.97 - 0.859i)T + (29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-10.1 - 2.99i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (3.27 - 0.962i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (0.788 + 0.113i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 0.573iT - 47T^{2} \) |
| 53 | \( 1 + (-2.71 - 3.12i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-5.33 + 6.16i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.26 - 8.77i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-0.841 - 1.30i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (5.17 + 8.04i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (0.463 - 1.01i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (3.93 - 4.53i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-2.87 + 9.79i)T + (-69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (7.00 + 1.00i)T + (85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (5.05 + 17.2i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−9.811359683645348596335882264176, −8.722584917737937159900492680804, −8.401050326833595329962210087139, −7.39968771569622607763732424298, −6.81480208329162459928355179707, −6.06017208643723017987767721160, −4.27628048445237293564435175725, −3.08032524191826606979357528887, −1.94860358604193465281464012101, −0.972853419209735019167475330006,
1.06712744713043167946202866513, 2.98827680076486291553832449457, 3.94178514557328067174965330774, 5.08587026045197729626870516820, 5.87730567333116741922033668660, 7.07003089641115883350009993432, 8.014433960190358281730198272093, 8.661427637732528239860322106711, 9.392292652955362902802454329685, 10.17442932655102884503979354919
