| L(s) = 1 | + (0.104 − 0.994i)2-s + (1.20 − 0.256i)3-s + (−0.978 − 0.207i)4-s + (2.60 + 1.16i)5-s + (−0.129 − 1.22i)6-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + (−1.34 + 0.598i)9-s + (1.42 − 2.47i)10-s + (0.309 − 3.30i)11-s − 1.23·12-s + (3.52 − 1.56i)13-s + (0.978 − 0.207i)14-s + (3.45 + 0.733i)15-s + (0.913 + 0.406i)16-s + (1.26 + 0.562i)17-s + ⋯ |
| L(s) = 1 | + (0.0739 − 0.703i)2-s + (0.698 − 0.148i)3-s + (−0.489 − 0.103i)4-s + (1.16 + 0.519i)5-s + (−0.0527 − 0.501i)6-s + (0.116 + 0.359i)7-s + (−0.109 + 0.336i)8-s + (−0.448 + 0.199i)9-s + (0.451 − 0.781i)10-s + (0.0931 − 0.995i)11-s − 0.356·12-s + (0.976 − 0.434i)13-s + (0.261 − 0.0555i)14-s + (0.890 + 0.189i)15-s + (0.228 + 0.101i)16-s + (0.306 + 0.136i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 + 0.688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.725 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.87162 - 0.746536i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.87162 - 0.746536i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.104 + 0.994i)T \) |
| 11 | \( 1 + (-0.309 + 3.30i)T \) |
| 19 | \( 1 + (-3.06 - 3.09i)T \) |
| good | 3 | \( 1 + (-1.20 + 0.256i)T + (2.74 - 1.22i)T^{2} \) |
| 5 | \( 1 + (-2.60 - 1.16i)T + (3.34 + 3.71i)T^{2} \) |
| 7 | \( 1 + (-0.309 - 0.951i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (-3.52 + 1.56i)T + (8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (-1.26 - 0.562i)T + (11.3 + 12.6i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.978 - 0.207i)T + (26.4 + 11.7i)T^{2} \) |
| 31 | \( 1 + (8.97 + 6.51i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.427 + 1.31i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (0.230 - 0.0490i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (1.26 + 2.18i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.669 + 0.743i)T + (-4.91 + 46.7i)T^{2} \) |
| 53 | \( 1 + (10.2 - 4.57i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (2.63 - 2.93i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-1.13 - 10.7i)T + (-59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (1.16 - 2.01i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (12.2 + 5.44i)T + (47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (7.32 - 8.13i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (-0.577 + 5.49i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (4.04 - 2.93i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-5.97 + 10.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.38 - 13.1i)T + (-94.8 - 20.1i)T^{2} \) |
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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
−11.00838124655866308833176079783, −10.27369384180291161624131604382, −9.265620941575879148773313466874, −8.619755058920051273778924825367, −7.63733475902311180757271122989, −5.89190617832276906399996991787, −5.66155731673512858222589725225, −3.63829989753968886330527450352, −2.80821128456800581078216083939, −1.66269674701012026002800966890,
1.67474052526989920447612679846, 3.31085117160440881185940631679, 4.63746910919622516375407752953, 5.60738416274530273775230824783, 6.59941151931882406880868687840, 7.62904478510246278282468043314, 8.773646106309826413276820333042, 9.261031923571704905916957541189, 10.01112647048443135452509284883, 11.25592451179392970740020679610
