| L(s) = 1 | + (−0.222 − 0.974i)2-s + (2.06 + 1.91i)3-s + (−0.900 + 0.433i)4-s + (−0.988 + 0.149i)5-s + (1.40 − 2.43i)6-s + (−0.0872 − 0.151i)7-s + (0.623 + 0.781i)8-s + (0.367 + 4.89i)9-s + (0.365 + 0.930i)10-s + (3.45 + 1.66i)11-s + (−2.68 − 0.829i)12-s + (−1.25 + 3.19i)13-s + (−0.127 + 0.118i)14-s + (−2.32 − 1.58i)15-s + (0.623 − 0.781i)16-s + (1.30 + 0.196i)17-s + ⋯ |
| L(s) = 1 | + (−0.157 − 0.689i)2-s + (1.19 + 1.10i)3-s + (−0.450 + 0.216i)4-s + (−0.442 + 0.0666i)5-s + (0.574 − 0.994i)6-s + (−0.0329 − 0.0571i)7-s + (0.220 + 0.276i)8-s + (0.122 + 1.63i)9-s + (0.115 + 0.294i)10-s + (1.04 + 0.501i)11-s + (−0.775 − 0.239i)12-s + (−0.347 + 0.886i)13-s + (−0.0342 + 0.0317i)14-s + (−0.600 − 0.409i)15-s + (0.155 − 0.195i)16-s + (0.316 + 0.0476i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.58262 + 0.592508i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.58262 + 0.592508i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.222 + 0.974i)T \) |
| 5 | \( 1 + (0.988 - 0.149i)T \) |
| 43 | \( 1 + (-2.31 + 6.13i)T \) |
| good | 3 | \( 1 + (-2.06 - 1.91i)T + (0.224 + 2.99i)T^{2} \) |
| 7 | \( 1 + (0.0872 + 0.151i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.45 - 1.66i)T + (6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (1.25 - 3.19i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-1.30 - 0.196i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (0.0478 - 0.638i)T + (-18.7 - 2.83i)T^{2} \) |
| 23 | \( 1 + (-2.61 + 1.78i)T + (8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (3.48 - 3.23i)T + (2.16 - 28.9i)T^{2} \) |
| 31 | \( 1 + (1.65 + 0.510i)T + (25.6 + 17.4i)T^{2} \) |
| 37 | \( 1 + (-0.805 + 1.39i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.795 - 3.48i)T + (-36.9 + 17.7i)T^{2} \) |
| 47 | \( 1 + (6.56 - 3.16i)T + (29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (2.60 + 6.64i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-5.47 + 6.87i)T + (-13.1 - 57.5i)T^{2} \) |
| 61 | \( 1 + (-2.74 + 0.848i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-0.109 + 1.45i)T + (-66.2 - 9.98i)T^{2} \) |
| 71 | \( 1 + (-0.335 - 0.228i)T + (25.9 + 66.0i)T^{2} \) |
| 73 | \( 1 + (-5.87 + 14.9i)T + (-53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (6.67 + 11.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.868 + 0.806i)T + (6.20 + 82.7i)T^{2} \) |
| 89 | \( 1 + (-10.9 - 10.1i)T + (6.65 + 88.7i)T^{2} \) |
| 97 | \( 1 + (16.8 + 8.09i)T + (60.4 + 75.8i)T^{2} \) |
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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
−11.13530711788274572781943346050, −10.18349673969938391738920212265, −9.393556177915571041116861034118, −8.938292108475094436069653370574, −7.940399256287783581200593651283, −6.83758662277533011692460584264, −4.92749219444951774633428139647, −4.04658634905442192737647461110, −3.35525741267753506060536961134, −1.99684349825378336642760916545,
1.13480934620082922648456432741, 2.84988384526535624949842730759, 3.95828601311910012694126241678, 5.59221402653591300207420211720, 6.72127411262744777998368442562, 7.48255417557402075381225144042, 8.166647879464230186241974212704, 8.922832974457714216644097906459, 9.704765535562853138040312715481, 11.16319304044093845880862721603
