| L(s) = 1 | − 5.05·2-s − 10.2·3-s + 17.6·4-s + 4.34·5-s + 51.6·6-s − 31.2·7-s − 48.5·8-s + 77.3·9-s − 21.9·10-s − 11·11-s − 179.·12-s + 158.·14-s − 44.3·15-s + 104.·16-s + 62.6·17-s − 391.·18-s + 49.4·19-s + 76.3·20-s + 319.·21-s + 55.6·22-s + 35.6·23-s + 496.·24-s − 106.·25-s − 514.·27-s − 550.·28-s + 148.·29-s + 224.·30-s + ⋯ |
| L(s) = 1 | − 1.78·2-s − 1.96·3-s + 2.20·4-s + 0.388·5-s + 3.51·6-s − 1.68·7-s − 2.14·8-s + 2.86·9-s − 0.694·10-s − 0.301·11-s − 4.32·12-s + 3.02·14-s − 0.763·15-s + 1.64·16-s + 0.894·17-s − 5.12·18-s + 0.596·19-s + 0.854·20-s + 3.31·21-s + 0.539·22-s + 0.322·23-s + 4.22·24-s − 0.849·25-s − 3.66·27-s − 3.71·28-s + 0.953·29-s + 1.36·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3332085038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3332085038\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 5.05T + 8T^{2} \) |
| 3 | \( 1 + 10.2T + 27T^{2} \) |
| 5 | \( 1 - 4.34T + 125T^{2} \) |
| 7 | \( 1 + 31.2T + 343T^{2} \) |
| 17 | \( 1 - 62.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 49.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 35.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 148.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 62.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 100.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 91.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 117.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 309.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 131.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 815.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 667.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 16.8T + 3.00e5T^{2} \) |
| 71 | \( 1 - 553.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 515.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 569.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 233.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 141.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.43e3T + 9.12e5T^{2} \) |
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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
−9.388584545212771618210255638967, −8.015596672039399527876865793695, −7.08892808582939199658189434662, −6.71466360703279002250445443338, −5.91441424188727485577831317314, −5.39487732189092604341701712311, −3.82288777293947786591654356627, −2.45073574928428306862085709275, −1.06573633355253555682397995696, −0.49704093264817287813424993915,
0.49704093264817287813424993915, 1.06573633355253555682397995696, 2.45073574928428306862085709275, 3.82288777293947786591654356627, 5.39487732189092604341701712311, 5.91441424188727485577831317314, 6.71466360703279002250445443338, 7.08892808582939199658189434662, 8.015596672039399527876865793695, 9.388584545212771618210255638967
