| L(s) = 1 | + i·2-s − 4-s − 5-s − i·8-s − i·10-s + 1.23i·11-s − 4.33i·13-s + 16-s − 4.12·17-s − 2.28i·19-s + 20-s − 1.23·22-s + 2.26i·23-s + 25-s + 4.33·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.447·5-s − 0.353i·8-s − 0.316i·10-s + 0.372i·11-s − 1.20i·13-s + 0.250·16-s − 1.00·17-s − 0.523i·19-s + 0.223·20-s − 0.263·22-s + 0.471i·23-s + 0.200·25-s + 0.849·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.062875083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.062875083\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 1.23iT - 11T^{2} \) |
| 13 | \( 1 + 4.33iT - 13T^{2} \) |
| 17 | \( 1 + 4.12T + 17T^{2} \) |
| 19 | \( 1 + 2.28iT - 19T^{2} \) |
| 23 | \( 1 - 2.26iT - 23T^{2} \) |
| 29 | \( 1 + 0.511iT - 29T^{2} \) |
| 31 | \( 1 - 3.89iT - 31T^{2} \) |
| 37 | \( 1 + 4.39T + 37T^{2} \) |
| 41 | \( 1 + 7.08T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 7.51T + 47T^{2} \) |
| 53 | \( 1 - 8.83iT - 53T^{2} \) |
| 59 | \( 1 - 8.33T + 59T^{2} \) |
| 61 | \( 1 + 0.890iT - 61T^{2} \) |
| 67 | \( 1 + 5.24T + 67T^{2} \) |
| 71 | \( 1 + 0.818iT - 71T^{2} \) |
| 73 | \( 1 - 11.6iT - 73T^{2} \) |
| 79 | \( 1 - 5.65T + 79T^{2} \) |
| 83 | \( 1 - 0.164T + 83T^{2} \) |
| 89 | \( 1 - 4.49T + 89T^{2} \) |
| 97 | \( 1 + 9.06iT - 97T^{2} \) |
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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
−8.627287278171162038810267913021, −7.68418383669644705255871701144, −7.25725067060268927103927314839, −6.50410087830564217846943702328, −5.64254081932527076456172456435, −4.97735131002728593414216499956, −4.20504562600232564477318945077, −3.35043303929357529423074999409, −2.35695287863784164822966327030, −0.881932334721022930931055384892,
0.38008234216554860548016116041, 1.72990662149074463586122804628, 2.51272607548319134652902993215, 3.61412193186716157003571869499, 4.18842551173284737855716452220, 4.91212848563591136050646787176, 5.91178442688325839656916781098, 6.70332669424874063347991427683, 7.41907349281074883353439442600, 8.370405339549437131241120201291
