| L(s) = 1 | + 3.31i·3-s + (0.5 − 4.97i)5-s − 2·9-s − 11·11-s + (16.5 + 1.65i)15-s − 29.8i·23-s + (−24.5 − 4.97i)25-s + 23.2i·27-s − 37·31-s − 36.4i·33-s − 69.6i·37-s + (−1 + 9.94i)45-s − 79.5i·47-s − 49·49-s + 79.5i·53-s + ⋯ |
| L(s) = 1 | + 1.10i·3-s + (0.100 − 0.994i)5-s − 0.222·9-s − 11-s + (1.10 + 0.110i)15-s − 1.29i·23-s + (−0.979 − 0.198i)25-s + 0.859i·27-s − 1.19·31-s − 1.10i·33-s − 1.88i·37-s + (−0.0222 + 0.221i)45-s − 1.69i·47-s − 0.999·49-s + 1.50i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0999 + 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0999 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8801176464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8801176464\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 4.97i)T \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 - 3.31iT - 9T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 + 29.8iT - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 37T + 961T^{2} \) |
| 37 | \( 1 + 69.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 + 79.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 79.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 107T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 + 129. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 133T + 5.04e3T^{2} \) |
| 73 | \( 1 + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 - 97T + 7.92e3T^{2} \) |
| 97 | \( 1 + 169. iT - 9.40e3T^{2} \) |
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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
−9.629854549652528725943928448110, −9.064588859605099193444908552480, −8.230908423020444085648290204254, −7.30396226144535399522362541194, −5.92250797889533402295577151855, −5.08548320711101022011783039080, −4.47181104282125413384741055626, −3.47886626337898939784039308646, −2.03522633239220612718141301653, −0.27824029883142625459962910771,
1.48892561888123075414068993020, 2.54127502441527476898082189550, 3.52677759539447497490955447897, 5.03869766389037898399216132765, 6.08422775764835938000296795384, 6.80669113419137349461071129576, 7.62737011397381522348605936218, 8.051624061049541142520817343035, 9.449532219929026204774930282909, 10.19439928940713143212467067916
