| L(s) = 1 | + (0.281 + 1.38i)2-s + (−0.209 + 2.83i)3-s + (−1.84 + 0.781i)4-s + (0.514 − 0.489i)5-s + (−3.99 + 0.510i)6-s + (0.0269 + 0.00674i)7-s + (−1.60 − 2.33i)8-s + (−5.05 − 0.749i)9-s + (0.823 + 0.574i)10-s + (1.42 + 5.14i)11-s + (−1.83 − 5.39i)12-s + (−6.23 − 2.40i)13-s + (−0.00175 + 0.0392i)14-s + (1.28 + 1.56i)15-s + (2.77 − 2.87i)16-s + (−1.22 − 1.00i)17-s + ⋯ |
| L(s) = 1 | + (0.199 + 0.979i)2-s + (−0.120 + 1.63i)3-s + (−0.920 + 0.390i)4-s + (0.229 − 0.218i)5-s + (−1.63 + 0.208i)6-s + (0.0101 + 0.00254i)7-s + (−0.566 − 0.824i)8-s + (−1.68 − 0.249i)9-s + (0.260 + 0.181i)10-s + (0.429 + 1.55i)11-s + (−0.529 − 1.55i)12-s + (−1.72 − 0.667i)13-s + (−0.000469 + 0.0104i)14-s + (0.331 + 0.403i)15-s + (0.694 − 0.719i)16-s + (−0.296 − 0.243i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.511 + 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.511 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.474574 - 0.835278i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.474574 - 0.835278i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.281 - 1.38i)T \) |
| good | 3 | \( 1 + (0.209 - 2.83i)T + (-2.96 - 0.440i)T^{2} \) |
| 5 | \( 1 + (-0.514 + 0.489i)T + (0.245 - 4.99i)T^{2} \) |
| 7 | \( 1 + (-0.0269 - 0.00674i)T + (6.17 + 3.29i)T^{2} \) |
| 11 | \( 1 + (-1.42 - 5.14i)T + (-9.43 + 5.65i)T^{2} \) |
| 13 | \( 1 + (6.23 + 2.40i)T + (9.63 + 8.73i)T^{2} \) |
| 17 | \( 1 + (1.22 + 1.00i)T + (3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (0.648 + 2.88i)T + (-17.1 + 8.12i)T^{2} \) |
| 23 | \( 1 + (0.868 - 2.42i)T + (-17.7 - 14.5i)T^{2} \) |
| 29 | \( 1 + (-4.48 - 3.49i)T + (7.04 + 28.1i)T^{2} \) |
| 31 | \( 1 + (2.48 + 0.493i)T + (28.6 + 11.8i)T^{2} \) |
| 37 | \( 1 + (-10.5 + 1.83i)T + (34.8 - 12.4i)T^{2} \) |
| 41 | \( 1 + (-0.418 - 8.51i)T + (-40.8 + 4.01i)T^{2} \) |
| 43 | \( 1 + (-9.74 - 8.40i)T + (6.30 + 42.5i)T^{2} \) |
| 47 | \( 1 + (4.57 + 8.56i)T + (-26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (1.24 - 0.975i)T + (12.8 - 51.4i)T^{2} \) |
| 59 | \( 1 + (-3.18 - 8.25i)T + (-43.7 + 39.6i)T^{2} \) |
| 61 | \( 1 + (4.65 - 1.53i)T + (48.9 - 36.3i)T^{2} \) |
| 67 | \( 1 + (5.10 + 2.57i)T + (39.9 + 53.8i)T^{2} \) |
| 71 | \( 1 + (2.96 + 2.19i)T + (20.6 + 67.9i)T^{2} \) |
| 73 | \( 1 + (7.05 - 1.76i)T + (64.3 - 34.4i)T^{2} \) |
| 79 | \( 1 + (-3.50 - 11.5i)T + (-65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (-2.61 - 0.453i)T + (78.1 + 27.9i)T^{2} \) |
| 89 | \( 1 + (0.962 + 2.68i)T + (-68.7 + 56.4i)T^{2} \) |
| 97 | \( 1 + (4.14 - 2.76i)T + (37.1 - 89.6i)T^{2} \) |
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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
−11.40347013434489497942645415743, −10.12546338096641714149577951171, −9.581741099953913928048816585188, −9.177503930812625151518005044288, −7.80591613282133949587827339480, −6.95345434761768084220863159619, −5.61637241717343210370969764934, −4.73453462518127093119864356526, −4.42577633993633063347606449224, −2.92336778785203722053704858146,
0.53185906083935392623171252923, 1.99394179976601569357829515570, 2.82756773442436412245703757145, 4.38990287185612513210650708820, 5.85661327036494394146687615737, 6.42306255863753217655344608876, 7.67446664952319283067084061390, 8.490419022675593173361049503812, 9.466262610831969335412196376159, 10.57319425234843994232661917226
