| L(s) = 1 | + (1.41 + i)3-s − 5-s + 2i·7-s + (1.00 + 2.82i)9-s − 2i·11-s + 2.82i·13-s + (−1.41 − i)15-s + 2.82i·17-s + 5.65·19-s + (−2 + 2.82i)21-s − 8.48·23-s + 25-s + (−1.41 + 5.00i)27-s + 2·29-s + 6i·31-s + ⋯ |
| L(s) = 1 | + (0.816 + 0.577i)3-s − 0.447·5-s + 0.755i·7-s + (0.333 + 0.942i)9-s − 0.603i·11-s + 0.784i·13-s + (−0.365 − 0.258i)15-s + 0.685i·17-s + 1.29·19-s + (−0.436 + 0.617i)21-s − 1.76·23-s + 0.200·25-s + (−0.272 + 0.962i)27-s + 0.371·29-s + 1.07i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.14177 + 1.35435i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14177 + 1.35435i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.41 - i)T \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 2.82iT - 13T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 6iT - 31T^{2} \) |
| 37 | \( 1 + 2.82iT - 37T^{2} \) |
| 41 | \( 1 - 11.3iT - 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + 10iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 2.82T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 2iT - 79T^{2} \) |
| 83 | \( 1 - 2iT - 83T^{2} \) |
| 89 | \( 1 - 5.65iT - 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−10.02475628735794592094955934029, −9.450860077807319241001612703084, −8.422518066967221270908245679010, −8.161963177954471796951062413679, −6.97651921962516698362626666600, −5.86702529935699244285319371130, −4.87969923338213709674395187584, −3.85774180610164283782017407454, −3.05328334719730069449245723287, −1.81745722432679392517838421018,
0.76152721493161220013518120062, 2.24032569123476888842818649292, 3.41293620261775500232256219069, 4.18804273383760282128669719657, 5.45440562769254213522818430190, 6.67673581016367996044427030021, 7.52982768800350001111641819368, 7.84284599335311756955865513116, 8.871642903694400161931694239100, 9.863979239325013398821815349076
