L(s) = 1 | + (1.73 + 1.73i)5-s − 3.86i·7-s + (3.34 + 3.34i)11-s + (−0.267 + 0.267i)13-s − 3.46·17-s + (3.34 − 3.34i)19-s − 1.79i·23-s + 0.999i·25-s + (1.73 − 1.73i)29-s + 5.65·31-s + (6.69 − 6.69i)35-s + (3.73 + 3.73i)37-s + 6.92i·41-s + (1.55 + 1.55i)43-s + 9.79·47-s + ⋯ |
L(s) = 1 | + (0.774 + 0.774i)5-s − 1.46i·7-s + (1.00 + 1.00i)11-s + (−0.0743 + 0.0743i)13-s − 0.840·17-s + (0.767 − 0.767i)19-s − 0.373i·23-s + 0.199i·25-s + (0.321 − 0.321i)29-s + 1.01·31-s + (1.13 − 1.13i)35-s + (0.613 + 0.613i)37-s + 1.08i·41-s + (0.236 + 0.236i)43-s + 1.42·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.244377570\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.244377570\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.73 - 1.73i)T + 5iT^{2} \) |
| 7 | \( 1 + 3.86iT - 7T^{2} \) |
| 11 | \( 1 + (-3.34 - 3.34i)T + 11iT^{2} \) |
| 13 | \( 1 + (0.267 - 0.267i)T - 13iT^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + (-3.34 + 3.34i)T - 19iT^{2} \) |
| 23 | \( 1 + 1.79iT - 23T^{2} \) |
| 29 | \( 1 + (-1.73 + 1.73i)T - 29iT^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + (-3.73 - 3.73i)T + 37iT^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 + (-1.55 - 1.55i)T + 43iT^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 + (7.73 + 7.73i)T + 53iT^{2} \) |
| 59 | \( 1 + (5.13 + 5.13i)T + 59iT^{2} \) |
| 61 | \( 1 + (-3.73 + 3.73i)T - 61iT^{2} \) |
| 67 | \( 1 + (4.38 - 4.38i)T - 67iT^{2} \) |
| 71 | \( 1 + 1.79iT - 71T^{2} \) |
| 73 | \( 1 - 2.53iT - 73T^{2} \) |
| 79 | \( 1 - 4.14T + 79T^{2} \) |
| 83 | \( 1 + (-1.55 + 1.55i)T - 83iT^{2} \) |
| 89 | \( 1 - 2.53iT - 89T^{2} \) |
| 97 | \( 1 - 3.46T + 97T^{2} \) |
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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
−9.256160104919845654463521682829, −8.096276392195172088327149650346, −7.20790859671745679158278275609, −6.70428775985764724976286239237, −6.21058723131358041475791106174, −4.72217423701857131688848893200, −4.30577769213393110231211331275, −3.14927464931236669965047322727, −2.14213633678754953676005849547, −0.965287679056874727432973979105,
1.09336151599985290382181379845, 2.12413680966248023924674570640, 3.11529152415550595831881730702, 4.25126030380717912895511574987, 5.31130278726225688371637277566, 5.83636015762279791002019263493, 6.36628807895224263958217948536, 7.59126646412815475058096566589, 8.589327107763077588228708302409, 9.067439983060554114068024991537