L(s) = 1 | + (1.19 + 0.763i)2-s + (0.834 + 1.81i)4-s + (−1.13 + 0.652i)5-s + (2.50 − 0.851i)7-s + (−0.394 + 2.80i)8-s + (−1.84 − 0.0861i)10-s + (1.66 + 0.960i)11-s − 4.88i·13-s + (3.63 + 0.898i)14-s + (−2.60 + 3.03i)16-s + (7.03 + 4.06i)17-s + (2.90 + 5.03i)19-s + (−2.12 − 1.51i)20-s + (1.24 + 2.41i)22-s + (−7.15 + 4.13i)23-s + ⋯ |
L(s) = 1 | + (0.841 + 0.539i)2-s + (0.417 + 0.908i)4-s + (−0.505 + 0.291i)5-s + (0.946 − 0.321i)7-s + (−0.139 + 0.990i)8-s + (−0.583 − 0.0272i)10-s + (0.501 + 0.289i)11-s − 1.35i·13-s + (0.970 + 0.240i)14-s + (−0.652 + 0.758i)16-s + (1.70 + 0.985i)17-s + (0.666 + 1.15i)19-s + (−0.476 − 0.337i)20-s + (0.265 + 0.514i)22-s + (−1.49 + 0.861i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.95423 + 1.65995i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.95423 + 1.65995i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.19 - 0.763i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.50 + 0.851i)T \) |
good | 5 | \( 1 + (1.13 - 0.652i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.66 - 0.960i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4.88iT - 13T^{2} \) |
| 17 | \( 1 + (-7.03 - 4.06i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.90 - 5.03i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (7.15 - 4.13i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 3.41T + 29T^{2} \) |
| 31 | \( 1 + (-0.682 + 1.18i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.49 + 9.52i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 4.57iT - 41T^{2} \) |
| 43 | \( 1 + 0.547iT - 43T^{2} \) |
| 47 | \( 1 + (-2.44 - 4.22i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.31 + 4.01i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.07 + 5.31i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-8.77 + 5.06i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.85 - 3.95i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4.66iT - 71T^{2} \) |
| 73 | \( 1 + (7.64 + 4.41i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.79 + 3.92i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 2.23T + 83T^{2} \) |
| 89 | \( 1 + (1.97 - 1.14i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 2.86iT - 97T^{2} \) |
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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
−10.66458622402569150348677536261, −9.834843094104746932311057771869, −8.276740045100936838645131173254, −7.78833588739874550852113053291, −7.29115462072351776606018599211, −5.68183349300999572305396306820, −5.51182225988629416610288729507, −3.82707845449730900264647571551, −3.61211282546393503567062435920, −1.75872676349866634658546530236,
1.15852053830735464777442949749, 2.46695567534948558195190018474, 3.77024977661583348838851945171, 4.63114497410612623551507515251, 5.38058909219484277136933029979, 6.49820990986693388932030507010, 7.46336066651104860594172920362, 8.502079880247426239036466802609, 9.443843080327834705214556978980, 10.26253000258072684922879584337