L(s) = 1 | + 3.88·3-s − 2.23i·5-s − 0.578i·7-s + 6.07·9-s + 9.86i·11-s − 10.3·13-s − 8.68i·15-s + 15.0i·17-s + 27.0i·19-s − 2.24i·21-s + (−20.5 + 10.2i)23-s − 5.00·25-s − 11.3·27-s − 19.2·29-s + 11.4·31-s + ⋯ |
L(s) = 1 | + 1.29·3-s − 0.447i·5-s − 0.0826i·7-s + 0.674·9-s + 0.896i·11-s − 0.792·13-s − 0.578i·15-s + 0.887i·17-s + 1.42i·19-s − 0.106i·21-s + (−0.895 + 0.445i)23-s − 0.200·25-s − 0.420·27-s − 0.664·29-s + 0.369·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.445 - 0.895i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.674131586\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.674131586\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (20.5 - 10.2i)T \) |
good | 3 | \( 1 - 3.88T + 9T^{2} \) |
| 7 | \( 1 + 0.578iT - 49T^{2} \) |
| 11 | \( 1 - 9.86iT - 121T^{2} \) |
| 13 | \( 1 + 10.3T + 169T^{2} \) |
| 17 | \( 1 - 15.0iT - 289T^{2} \) |
| 19 | \( 1 - 27.0iT - 361T^{2} \) |
| 29 | \( 1 + 19.2T + 841T^{2} \) |
| 31 | \( 1 - 11.4T + 961T^{2} \) |
| 37 | \( 1 + 28.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 60.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + 4.51iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 68.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 54.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 73.8T + 3.48e3T^{2} \) |
| 61 | \( 1 - 40.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 9.50iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 82.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 71.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 132. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 74.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 123. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 184. iT - 9.40e3T^{2} \) |
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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
−9.286803717337373620582040552556, −8.465205561503410956385533383295, −7.86550915568032649817208440366, −7.30443195028119462337270256846, −6.13180235904362057860406146716, −5.19007210022391467340513532080, −4.10686503533490714874702922029, −3.54022387204655591909557960763, −2.26005314974825784821385199517, −1.64330925080603233335252440668,
0.31913926935956890643353934648, 2.06005057063247504363799641285, 2.83247007835675610638952985032, 3.44783728133144646358483444722, 4.58870514571496657182231870810, 5.53127126791770172612837340044, 6.70923826007335023498037184325, 7.26280130885595542454250160201, 8.296645853559482481967308794530, 8.590269355793028042702626436394