| L(s) = 1 | − 43.1i·5-s + (44.4 + 20.6i)7-s − 11.8·11-s + 20.6i·13-s + 289. i·17-s + 104. i·19-s + 73.5·23-s − 1.23e3·25-s − 950.·29-s − 1.38e3i·31-s + (889. − 1.91e3i)35-s − 1.27e3·37-s − 1.30e3i·41-s + 96.2·43-s + 186. i·47-s + ⋯ |
| L(s) = 1 | − 1.72i·5-s + (0.907 + 0.420i)7-s − 0.0977·11-s + 0.121i·13-s + 1.00i·17-s + 0.289i·19-s + 0.138·23-s − 1.97·25-s − 1.13·29-s − 1.44i·31-s + (0.725 − 1.56i)35-s − 0.934·37-s − 0.775i·41-s + 0.0520·43-s + 0.0842i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4016844247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4016844247\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-44.4 - 20.6i)T \) |
| good | 5 | \( 1 + 43.1iT - 625T^{2} \) |
| 11 | \( 1 + 11.8T + 1.46e4T^{2} \) |
| 13 | \( 1 - 20.6iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 289. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 104. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 73.5T + 2.79e5T^{2} \) |
| 29 | \( 1 + 950.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.38e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.27e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.30e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 96.2T + 3.41e6T^{2} \) |
| 47 | \( 1 - 186. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 4.37e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 1.65e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 5.20e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 552.T + 2.01e7T^{2} \) |
| 71 | \( 1 + 8.48e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 317. iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 624.T + 3.89e7T^{2} \) |
| 83 | \( 1 + 7.66e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 4.19e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.29e4iT - 8.85e7T^{2} \) |
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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
−8.901992812010485699740110172245, −8.180293380853681175954209248310, −7.60216759106220929295235072963, −6.06471092490913707466935700385, −5.38088037737138851523631271602, −4.61748923059011449269935451051, −3.79730747001530941132253367020, −2.03542649273996939836356765002, −1.35636001800642175874382403610, −0.080568496996156241856022069767,
1.54322544446528068372015428619, 2.69920880378162922127963453477, 3.46116376858927497139438197541, 4.64449746686938718421299711824, 5.62598927470281982698002759991, 6.77668852855800707079650841422, 7.23859923090620014530890144607, 7.990676628286875085181910142033, 9.101673281130194476690901120795, 10.10697780194590225315761603275
