L(s) = 1 | + (−0.816 − 1.52i)3-s − 1.78·5-s + (1.90 + 1.83i)7-s + (−1.66 + 2.49i)9-s − 5.61·11-s + (3.14 + 5.43i)13-s + (1.45 + 2.72i)15-s + (0.646 + 1.11i)17-s + (0.559 − 0.968i)19-s + (1.25 − 4.40i)21-s + 7.61·23-s − 1.81·25-s + (5.17 + 0.507i)27-s + (−1.57 + 2.72i)29-s + (−0.501 + 0.868i)31-s + ⋯ |
L(s) = 1 | + (−0.471 − 0.881i)3-s − 0.797·5-s + (0.718 + 0.695i)7-s + (−0.555 + 0.831i)9-s − 1.69·11-s + (0.870 + 1.50i)13-s + (0.376 + 0.703i)15-s + (0.156 + 0.271i)17-s + (0.128 − 0.222i)19-s + (0.274 − 0.961i)21-s + 1.58·23-s − 0.363·25-s + (0.995 + 0.0977i)27-s + (−0.292 + 0.506i)29-s + (−0.0900 + 0.156i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.409 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.653921 + 0.423395i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.653921 + 0.423395i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.816 + 1.52i)T \) |
| 7 | \( 1 + (-1.90 - 1.83i)T \) |
good | 5 | \( 1 + 1.78T + 5T^{2} \) |
| 11 | \( 1 + 5.61T + 11T^{2} \) |
| 13 | \( 1 + (-3.14 - 5.43i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.646 - 1.11i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.559 + 0.968i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 7.61T + 23T^{2} \) |
| 29 | \( 1 + (1.57 - 2.72i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.501 - 0.868i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.96 - 10.3i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.14 - 7.17i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.34 + 4.06i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.972 - 1.68i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.45 + 7.72i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.19 - 7.26i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.41 + 4.17i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.27 + 2.21i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8.86T + 71T^{2} \) |
| 73 | \( 1 + (-5.67 - 9.83i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.72 + 11.6i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.60 - 2.77i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.404 - 0.700i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.10 + 1.91i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.31443775834954631296041025410, −10.58248934914323101921967139086, −8.990525473001052854344324683621, −8.240552547552207728057575406605, −7.53126580178434770315719778381, −6.58149776636005035717019721426, −5.43284609491052813145673222227, −4.64762997229404063179519588217, −2.94544470837748336980983702454, −1.59493931833482601741665417240,
0.50922066765189818902212596094, 3.03383664878383254448965098064, 4.00635348681558106473322689775, 5.10794698507455415125399878138, 5.71795845064231756823405033261, 7.42775562919790981752218701370, 7.932684585349227160897669598391, 8.951933901152635586573292548722, 10.25252355019612767544088858599, 10.82392587778453238570509987641