L(s) = 1 | + (0.416 + 0.817i)2-s + (−0.987 + 0.156i)3-s + (0.681 − 0.937i)4-s + (−1.99 + 1.00i)5-s + (−0.539 − 0.741i)6-s + (−1.95 − 1.78i)7-s + (2.86 + 0.453i)8-s + (0.951 − 0.309i)9-s + (−1.65 − 1.21i)10-s + (0.0792 − 0.244i)11-s + (−0.526 + 1.03i)12-s + (4.71 + 2.40i)13-s + (0.641 − 2.33i)14-s + (1.81 − 1.30i)15-s + (0.104 + 0.322i)16-s + (7.72 + 1.22i)17-s + ⋯ |
L(s) = 1 | + (0.294 + 0.577i)2-s + (−0.570 + 0.0903i)3-s + (0.340 − 0.468i)4-s + (−0.893 + 0.448i)5-s + (−0.220 − 0.302i)6-s + (−0.739 − 0.673i)7-s + (1.01 + 0.160i)8-s + (0.317 − 0.103i)9-s + (−0.522 − 0.384i)10-s + (0.0239 − 0.0735i)11-s + (−0.151 + 0.298i)12-s + (1.30 + 0.665i)13-s + (0.171 − 0.625i)14-s + (0.469 − 0.336i)15-s + (0.0261 + 0.0805i)16-s + (1.87 + 0.296i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.39797 + 0.201022i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.39797 + 0.201022i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.987 - 0.156i)T \) |
| 5 | \( 1 + (1.99 - 1.00i)T \) |
| 7 | \( 1 + (1.95 + 1.78i)T \) |
good | 2 | \( 1 + (-0.416 - 0.817i)T + (-1.17 + 1.61i)T^{2} \) |
| 11 | \( 1 + (-0.0792 + 0.244i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-4.71 - 2.40i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-7.72 - 1.22i)T + (16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-3.91 + 2.84i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.78 + 0.907i)T + (13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (3.94 - 5.43i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (6.53 + 8.99i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.47 - 1.26i)T + (21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (-5.64 + 1.83i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (0.714 + 0.714i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.700 - 4.42i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (0.792 + 5.00i)T + (-50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (-1.71 - 5.27i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (10.2 + 3.32i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (2.07 - 13.1i)T + (-63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (3.96 + 2.88i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.213 - 0.419i)T + (-42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-7.45 + 10.2i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (1.25 - 7.90i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (1.75 - 5.39i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.741 - 4.67i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−11.02690435719634170232538124097, −10.22792775046667299706827354422, −9.248763926313728674007780609674, −7.69387654724184585264535592745, −7.22207301777708723157210719635, −6.28494118647699125615658925268, −5.55263695950474202733293020536, −4.21652833975226729827459503028, −3.34329777192273982457451402980, −1.07185448370486512785163045467,
1.23971720822237077260998095116, 3.21359658200882497645606010929, 3.68220218762076752869487991121, 5.18921660722572833091931054503, 6.03200711207305914266058475151, 7.39414750419121756425519697126, 7.933350704043619140368114425719, 9.095482740048372048469876850453, 10.19332768813096609441786963101, 11.10143717785060677224040693514