| L(s) = 1 | + 1.48i·3-s − 4.43·5-s − 2.39·7-s + 6.80·9-s + 12.1·11-s + 3.28i·13-s − 6.57i·15-s − 7.93·17-s + (18.4 − 4.71i)19-s − 3.54i·21-s − 22.6·23-s − 5.33·25-s + 23.4i·27-s + 42.2i·29-s + 6.86i·31-s + ⋯ |
| L(s) = 1 | + 0.494i·3-s − 0.887·5-s − 0.341·7-s + 0.755·9-s + 1.10·11-s + 0.252i·13-s − 0.438i·15-s − 0.466·17-s + (0.968 − 0.248i)19-s − 0.168i·21-s − 0.985·23-s − 0.213·25-s + 0.867i·27-s + 1.45i·29-s + 0.221i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 - 0.968i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.267478154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.267478154\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-18.4 + 4.71i)T \) |
| good | 3 | \( 1 - 1.48iT - 9T^{2} \) |
| 5 | \( 1 + 4.43T + 25T^{2} \) |
| 7 | \( 1 + 2.39T + 49T^{2} \) |
| 11 | \( 1 - 12.1T + 121T^{2} \) |
| 13 | \( 1 - 3.28iT - 169T^{2} \) |
| 17 | \( 1 + 7.93T + 289T^{2} \) |
| 23 | \( 1 + 22.6T + 529T^{2} \) |
| 29 | \( 1 - 42.2iT - 841T^{2} \) |
| 31 | \( 1 - 6.86iT - 961T^{2} \) |
| 37 | \( 1 - 40.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 59.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.93T + 1.84e3T^{2} \) |
| 47 | \( 1 - 12.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 89.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 51.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 14.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 74.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 88.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 8.33T + 5.32e3T^{2} \) |
| 79 | \( 1 + 7.24iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 55.6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 11.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 20.8iT - 9.40e3T^{2} \) |
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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
−10.69059506506444174803436819855, −9.712375638929093633424258129484, −9.142491789069016945109802910883, −8.040670908066725364546575359746, −7.11551768922271361363390755322, −6.34088960083639821709628359158, −4.88174211268247061313148320212, −4.07246619175203240204032511698, −3.26058539348311106367074380940, −1.37198133088595991435255823281,
0.51963538450747291883232204418, 1.99865985893481326120369677561, 3.66586303707008178684698063886, 4.25510652568701711861020716336, 5.77507271325719062396131653603, 6.74659545311648512964131749368, 7.50397581438897112803937436586, 8.235233484626881230276565647605, 9.397290199968407616913580028353, 10.06692122469431409739014707101
