| L(s) = 1 | + (−0.851 + 0.523i)2-s + (0.999 − 2.66i)3-s + (0.451 − 0.892i)4-s + (−2.22 − 3.03i)5-s + (0.546 + 2.79i)6-s + (0.329 − 0.504i)7-s + (0.0825 + 0.996i)8-s + (−3.86 − 3.36i)9-s + (3.48 + 1.41i)10-s + (−1.53 − 0.525i)11-s + (−1.93 − 2.09i)12-s + (−0.352 − 0.427i)13-s + (−0.0166 + 0.602i)14-s + (−10.3 + 2.92i)15-s + (−0.592 − 0.805i)16-s + (0.0653 + 0.129i)17-s + ⋯ |
| L(s) = 1 | + (−0.602 + 0.370i)2-s + (0.577 − 1.54i)3-s + (0.225 − 0.446i)4-s + (−0.997 − 1.35i)5-s + (0.223 + 1.14i)6-s + (0.124 − 0.190i)7-s + (0.0291 + 0.352i)8-s + (−1.28 − 1.12i)9-s + (1.10 + 0.448i)10-s + (−0.461 − 0.158i)11-s + (−0.557 − 0.605i)12-s + (−0.0976 − 0.118i)13-s + (−0.00443 + 0.161i)14-s + (−2.66 + 0.754i)15-s + (−0.148 − 0.201i)16-s + (0.0158 + 0.0313i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0751638 + 0.838881i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0751638 + 0.838881i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.851 - 0.523i)T \) |
| 19 | \( 1 + (-4.29 + 0.769i)T \) |
| good | 3 | \( 1 + (-0.999 + 2.66i)T + (-2.26 - 1.97i)T^{2} \) |
| 5 | \( 1 + (2.22 + 3.03i)T + (-1.49 + 4.77i)T^{2} \) |
| 7 | \( 1 + (-0.329 + 0.504i)T + (-2.81 - 6.41i)T^{2} \) |
| 11 | \( 1 + (1.53 + 0.525i)T + (8.68 + 6.75i)T^{2} \) |
| 13 | \( 1 + (0.352 + 0.427i)T + (-2.49 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-0.0653 - 0.129i)T + (-10.0 + 13.6i)T^{2} \) |
| 23 | \( 1 + (0.881 + 2.35i)T + (-17.3 + 15.1i)T^{2} \) |
| 29 | \( 1 + (-3.11 - 0.172i)T + (28.8 + 3.19i)T^{2} \) |
| 31 | \( 1 + (1.63 - 0.884i)T + (16.9 - 25.9i)T^{2} \) |
| 37 | \( 1 + (-9.82 - 3.37i)T + (29.1 + 22.7i)T^{2} \) |
| 41 | \( 1 + (6.24 + 6.07i)T + (1.12 + 40.9i)T^{2} \) |
| 43 | \( 1 + (7.06 - 2.86i)T + (30.8 - 29.9i)T^{2} \) |
| 47 | \( 1 + (0.912 + 4.67i)T + (-43.5 + 17.6i)T^{2} \) |
| 53 | \( 1 + (-1.96 - 10.0i)T + (-49.1 + 19.9i)T^{2} \) |
| 59 | \( 1 + (1.17 + 1.13i)T + (1.62 + 58.9i)T^{2} \) |
| 61 | \( 1 + (2.83 - 1.33i)T + (38.7 - 47.0i)T^{2} \) |
| 67 | \( 1 + (0.0366 - 0.0253i)T + (23.4 - 62.7i)T^{2} \) |
| 71 | \( 1 + (-3.50 - 1.65i)T + (45.1 + 54.8i)T^{2} \) |
| 73 | \( 1 + (6.38 + 12.6i)T + (-43.2 + 58.8i)T^{2} \) |
| 79 | \( 1 + (-12.3 + 5.00i)T + (56.6 - 55.0i)T^{2} \) |
| 83 | \( 1 + (0.865 + 1.97i)T + (-56.2 + 61.0i)T^{2} \) |
| 89 | \( 1 + (2.38 - 4.70i)T + (-52.7 - 71.7i)T^{2} \) |
| 97 | \( 1 + (5.57 + 3.86i)T + (34.0 + 90.8i)T^{2} \) |
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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
−9.556487708095751256318625234404, −8.677362092432236572181520878987, −8.112496363717462710580071251694, −7.63570658028610383371871232275, −6.82681823331909130569237642401, −5.63566273651137808265520201116, −4.54244929558080563630766878534, −2.99136684885020033983005059984, −1.48901816997935041765399328155, −0.51563475961527527360949298724,
2.55719501771061277134093794235, 3.32937192272341140152395566196, 4.04200585488667919998206420103, 5.20920637503979918269189301284, 6.72492510054917468501444915880, 7.76244207976380906464009308821, 8.287446169567388480241533982190, 9.454364356762192868730237642183, 9.997637390890679058760608474164, 10.68894573917524286702485179466
