| L(s) = 1 | − 3.46i·5-s − 3.46·7-s − 6·17-s + 4i·19-s − 6.92·23-s − 6.99·25-s − 3.46i·29-s + 3.46·31-s + 11.9i·35-s − 6.92i·37-s + 6·41-s − 4i·43-s − 6.92·47-s + 4.99·49-s + 3.46i·53-s + ⋯ |
| L(s) = 1 | − 1.54i·5-s − 1.30·7-s − 1.45·17-s + 0.917i·19-s − 1.44·23-s − 1.39·25-s − 0.643i·29-s + 0.622·31-s + 2.02i·35-s − 1.13i·37-s + 0.937·41-s − 0.609i·43-s − 1.01·47-s + 0.714·49-s + 0.475i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0714142 - 0.542445i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0714142 - 0.542445i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 + 3.46iT - 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 + 6.92iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 - 3.46iT - 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−10.02989057263215053938642930455, −9.457353418372534879309284036392, −8.656571848222829156585026925487, −7.84507612092177261103405542132, −6.49884461459931999062935756134, −5.78634424990708311522391769652, −4.57785343525255765807176907092, −3.73345116502139062215289856543, −2.07069782043318117443728681898, −0.28700099775010193211999135644,
2.45087938671556506621215799675, 3.21163461637970297056113286163, 4.37100312099716250778362272705, 6.04917201706116232523680282273, 6.61726129034001713973828793813, 7.24922420199386008717698723083, 8.531454437995464666502181061694, 9.618582323950146814403088677850, 10.21986442940077278589817239545, 11.04976536597921975684010107815
