L(s) = 1 | + (−0.344 + 1.05i)3-s + (−0.665 + 0.483i)5-s + (0.816 + 2.51i)7-s + (1.42 + 1.03i)9-s + (−3.22 − 0.761i)11-s + (0.809 + 0.587i)13-s + (−0.283 − 0.871i)15-s + (−4.38 + 3.18i)17-s + (1.81 − 5.59i)19-s − 2.94·21-s − 2.27·23-s + (−1.33 + 4.11i)25-s + (−4.28 + 3.11i)27-s + (2.34 + 7.22i)29-s + (−0.936 − 0.680i)31-s + ⋯ |
L(s) = 1 | + (−0.198 + 0.611i)3-s + (−0.297 + 0.216i)5-s + (0.308 + 0.949i)7-s + (0.474 + 0.344i)9-s + (−0.973 − 0.229i)11-s + (0.224 + 0.163i)13-s + (−0.0730 − 0.224i)15-s + (−1.06 + 0.773i)17-s + (0.417 − 1.28i)19-s − 0.642·21-s − 0.473·23-s + (−0.267 + 0.822i)25-s + (−0.825 + 0.599i)27-s + (0.436 + 1.34i)29-s + (−0.168 − 0.122i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.670 - 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.670 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.416270 + 0.937001i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.416270 + 0.937001i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + (3.22 + 0.761i)T \) |
| 13 | \( 1 + (-0.809 - 0.587i)T \) |
good | 3 | \( 1 + (0.344 - 1.05i)T + (-2.42 - 1.76i)T^{2} \) |
| 5 | \( 1 + (0.665 - 0.483i)T + (1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (-0.816 - 2.51i)T + (-5.66 + 4.11i)T^{2} \) |
| 17 | \( 1 + (4.38 - 3.18i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.81 + 5.59i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 2.27T + 23T^{2} \) |
| 29 | \( 1 + (-2.34 - 7.22i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.936 + 0.680i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.433 - 1.33i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.26 - 3.89i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 2.44T + 43T^{2} \) |
| 47 | \( 1 + (1.84 - 5.67i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (7.23 + 5.25i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.80 - 5.55i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-8.67 + 6.30i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 3.00T + 67T^{2} \) |
| 71 | \( 1 + (-7.89 + 5.73i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.32 - 4.07i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (11.2 + 8.14i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-3.92 + 2.85i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + (-2.15 - 1.56i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.07330204664195582768876286874, −10.31686378037707922591723104145, −9.282005047767800887281780119997, −8.510499030700655315636892357380, −7.56891838013486034176110624239, −6.48526796897033237110633922757, −5.28900466773451541730783662839, −4.67923459866239840745968581059, −3.34784889193230842036701418907, −2.05605701684000618770552752674,
0.58741949172446574308107862914, 2.11318418241383746355944106599, 3.80464778210182708453555407066, 4.66421497415565389488933544303, 5.92878026456759296195375507843, 6.95493914043655659215481489368, 7.69102624282009073958049287253, 8.347032766854376056855515446195, 9.772207298632430202609459851008, 10.33966595829073570919151063819