| L(s) = 1 | + 5.85i·3-s + 5.78·5-s − 25.2·9-s + 3.01i·11-s − 9.78·13-s + 33.8i·15-s + 11.6·17-s + 25.5i·19-s + 26.1i·23-s + 8.42·25-s − 95.4i·27-s + 1.56·29-s + 12.0i·31-s − 17.6·33-s − 70.6·37-s + ⋯ |
| L(s) = 1 | + 1.95i·3-s + 1.15·5-s − 2.81·9-s + 0.274i·11-s − 0.752·13-s + 2.25i·15-s + 0.682·17-s + 1.34i·19-s + 1.13i·23-s + 0.337·25-s − 3.53i·27-s + 0.0539·29-s + 0.387i·31-s − 0.535·33-s − 1.91·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.316482138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.316482138\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 5.85iT - 9T^{2} \) |
| 5 | \( 1 - 5.78T + 25T^{2} \) |
| 11 | \( 1 - 3.01iT - 121T^{2} \) |
| 13 | \( 1 + 9.78T + 169T^{2} \) |
| 17 | \( 1 - 11.6T + 289T^{2} \) |
| 19 | \( 1 - 25.5iT - 361T^{2} \) |
| 23 | \( 1 - 26.1iT - 529T^{2} \) |
| 29 | \( 1 - 1.56T + 841T^{2} \) |
| 31 | \( 1 - 12.0iT - 961T^{2} \) |
| 37 | \( 1 + 70.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 49.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 73.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 44.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 54.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 12.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 35.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 24.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 11.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 74.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 22.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 48.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 67.4T + 7.92e3T^{2} \) |
| 97 | \( 1 - 7.75T + 9.40e3T^{2} \) |
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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
−9.773055590977475468404613515036, −9.324646282923296821224035711412, −8.485667793018291428034931487930, −7.44277236349575961414406957029, −6.01729237247931599886145392358, −5.52900106126182043556753462439, −4.86954026239383804352598612984, −3.83893678375329971270484855786, −3.07072707132516570379156595000, −1.86124752431106588865745721449,
0.33008459489216024889924496834, 1.42902165161689550834272896776, 2.34704326467979468059406563266, 2.97711784132321726168519998983, 4.91238689361908565989829479624, 5.76208435279903281876094797571, 6.39666850243405155235901069577, 7.06712679674676561629791225853, 7.78540885922440205174666176991, 8.658445401078225336437101808346
