L(s) = 1 | + (−0.415 − 0.909i)2-s + (0.797 − 0.234i)3-s + (−0.654 + 0.755i)4-s + (0.841 + 0.540i)5-s + (−0.544 − 0.627i)6-s + (−0.0405 − 0.281i)7-s + (0.959 + 0.281i)8-s + (−0.260 + 0.167i)9-s + (0.142 − 0.989i)10-s + (−0.345 + 0.755i)12-s + (−0.239 + 0.153i)14-s + (0.797 + 0.234i)15-s + (−0.142 − 0.989i)16-s + (0.260 + 0.167i)18-s + (−0.959 + 0.281i)20-s + (−0.0982 − 0.215i)21-s + ⋯ |
L(s) = 1 | + (−0.415 − 0.909i)2-s + (0.797 − 0.234i)3-s + (−0.654 + 0.755i)4-s + (0.841 + 0.540i)5-s + (−0.544 − 0.627i)6-s + (−0.0405 − 0.281i)7-s + (0.959 + 0.281i)8-s + (−0.260 + 0.167i)9-s + (0.142 − 0.989i)10-s + (−0.345 + 0.755i)12-s + (−0.239 + 0.153i)14-s + (0.797 + 0.234i)15-s + (−0.142 − 0.989i)16-s + (0.260 + 0.167i)18-s + (−0.959 + 0.281i)20-s + (−0.0982 − 0.215i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9156207480\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9156207480\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.415 + 0.909i)T \) |
| 5 | \( 1 + (-0.841 - 0.540i)T \) |
| 23 | \( 1 + (0.415 + 0.909i)T \) |
good | 3 | \( 1 + (-0.797 + 0.234i)T + (0.841 - 0.540i)T^{2} \) |
| 7 | \( 1 + (0.0405 + 0.281i)T + (-0.959 + 0.281i)T^{2} \) |
| 11 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 13 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 17 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 19 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 29 | \( 1 + (1.10 + 1.27i)T + (-0.142 + 0.989i)T^{2} \) |
| 31 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 37 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 41 | \( 1 + (0.239 + 0.153i)T + (0.415 + 0.909i)T^{2} \) |
| 43 | \( 1 + (1.25 - 0.368i)T + (0.841 - 0.540i)T^{2} \) |
| 47 | \( 1 + 1.68T + T^{2} \) |
| 53 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 59 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 61 | \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \) |
| 67 | \( 1 + (-0.797 - 1.74i)T + (-0.654 + 0.755i)T^{2} \) |
| 71 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 73 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 79 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 83 | \( 1 + (-1.61 + 1.03i)T + (0.415 - 0.909i)T^{2} \) |
| 89 | \( 1 + (0.797 - 0.234i)T + (0.841 - 0.540i)T^{2} \) |
| 97 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
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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
−11.03681308059800758543969283493, −10.15681809447826327542706096642, −9.519306006638442336569401208722, −8.563145588480550003253698761997, −7.81410735549305902474322566865, −6.72375255795949475246923912138, −5.34875749239491039102613365839, −3.87099769901459644457773296430, −2.76637376910511324667357269487, −1.91352392781156901862574849936,
1.85253555252842241386988273216, 3.55227430695356547925973151471, 5.01244197742317207939267254419, 5.78034273270603811153484906285, 6.79910417142291441246842416073, 8.028251359377525447111527967943, 8.706489693949095080765070779187, 9.438660940717777525934566087365, 9.947452276459989187369494191557, 11.19298607653153340266802678199