L(s) = 1 | + (−1.78 − 0.713i)2-s + (1.94 + 1.85i)4-s + (−1.34 − 2.93i)8-s + (0.981 − 0.189i)9-s + (0.771 − 0.308i)11-s + (0.167 + 3.50i)16-s + (−1.88 − 0.363i)18-s − 1.59·22-s + (−0.327 − 0.945i)23-s + (−0.786 + 0.618i)25-s + (0.273 + 0.0801i)29-s + (1.14 − 3.32i)32-s + (2.25 + 1.45i)36-s + (1.65 − 0.318i)37-s + (−0.544 + 1.19i)43-s + (2.06 + 0.828i)44-s + ⋯ |
L(s) = 1 | + (−1.78 − 0.713i)2-s + (1.94 + 1.85i)4-s + (−1.34 − 2.93i)8-s + (0.981 − 0.189i)9-s + (0.771 − 0.308i)11-s + (0.167 + 3.50i)16-s + (−1.88 − 0.363i)18-s − 1.59·22-s + (−0.327 − 0.945i)23-s + (−0.786 + 0.618i)25-s + (0.273 + 0.0801i)29-s + (1.14 − 3.32i)32-s + (2.25 + 1.45i)36-s + (1.65 − 0.318i)37-s + (−0.544 + 1.19i)43-s + (2.06 + 0.828i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5240252371\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5240252371\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 23 | \( 1 + (0.327 + 0.945i)T \) |
good | 2 | \( 1 + (1.78 + 0.713i)T + (0.723 + 0.690i)T^{2} \) |
| 3 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 5 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 11 | \( 1 + (-0.771 + 0.308i)T + (0.723 - 0.690i)T^{2} \) |
| 13 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 17 | \( 1 + (0.888 - 0.458i)T^{2} \) |
| 19 | \( 1 + (0.888 + 0.458i)T^{2} \) |
| 29 | \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \) |
| 31 | \( 1 + (0.327 + 0.945i)T^{2} \) |
| 37 | \( 1 + (-1.65 + 0.318i)T + (0.928 - 0.371i)T^{2} \) |
| 41 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 43 | \( 1 + (0.544 - 1.19i)T + (-0.654 - 0.755i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.252 - 0.130i)T + (0.580 + 0.814i)T^{2} \) |
| 59 | \( 1 + (0.995 + 0.0950i)T^{2} \) |
| 61 | \( 1 + (-0.981 - 0.189i)T^{2} \) |
| 67 | \( 1 + (-1.50 + 1.18i)T + (0.235 - 0.971i)T^{2} \) |
| 71 | \( 1 + (0.118 + 0.822i)T + (-0.959 + 0.281i)T^{2} \) |
| 73 | \( 1 + (-0.0475 - 0.998i)T^{2} \) |
| 79 | \( 1 + (-0.252 + 0.130i)T + (0.580 - 0.814i)T^{2} \) |
| 83 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 89 | \( 1 + (0.327 - 0.945i)T^{2} \) |
| 97 | \( 1 + (0.142 + 0.989i)T^{2} \) |
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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
−9.635422470322342298643234316209, −9.432223485688093580567907747646, −8.377404708513062911600608201723, −7.75285274015612189776583228334, −6.87026208858937101552438070450, −6.19053536935441266448918712079, −4.32406654422176469176113533216, −3.36140842843713122473140111641, −2.13503069472381162042096921836, −1.06338571190988432946504560913,
1.23600636111698756311311627262, 2.25722802809866430814011184894, 4.07060767481808551899350581364, 5.41048462070235961124002868466, 6.37991516445185632423747939096, 7.02347794785143534536035596263, 7.76144178148624569814708276042, 8.435588739579030824801620468890, 9.457089058638790907327794120595, 9.804055547268963343124224065990