| L(s) = 1 | + (−4.59 + 2.43i)3-s + (1.72 + 2.99i)5-s + (−17.7 + 5.23i)7-s + (15.1 − 22.3i)9-s + (−26.9 − 15.5i)11-s + (−2.16 − 1.24i)13-s + (−15.2 − 9.53i)15-s + (0.621 + 1.07i)17-s + (76.0 + 43.8i)19-s + (68.7 − 67.2i)21-s + (−43.0 + 24.8i)23-s + (56.5 − 97.8i)25-s + (−15.1 + 139. i)27-s + (228. − 132. i)29-s − 257. i·31-s + ⋯ |
| L(s) = 1 | + (−0.883 + 0.468i)3-s + (0.154 + 0.267i)5-s + (−0.959 + 0.282i)7-s + (0.560 − 0.827i)9-s + (−0.738 − 0.426i)11-s + (−0.0461 − 0.0266i)13-s + (−0.262 − 0.164i)15-s + (0.00886 + 0.0153i)17-s + (0.918 + 0.530i)19-s + (0.714 − 0.699i)21-s + (−0.390 + 0.225i)23-s + (0.452 − 0.783i)25-s + (−0.107 + 0.994i)27-s + (1.46 − 0.845i)29-s − 1.48i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8573810336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8573810336\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.59 - 2.43i)T \) |
| 7 | \( 1 + (17.7 - 5.23i)T \) |
| good | 5 | \( 1 + (-1.72 - 2.99i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (26.9 + 15.5i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (2.16 + 1.24i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-0.621 - 1.07i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-76.0 - 43.8i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (43.0 - 24.8i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-228. + 132. i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + 257. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (47.7 - 82.6i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-93.9 + 162. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (133. + 230. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 - 571.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-167. + 96.8i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 - 291.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 158. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 443.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 405. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (505. - 291. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + 1.28e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + (610. + 1.05e3i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (406. - 704. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-496. + 286. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.54782270183253876741523405533, −10.25244483622218225875265949572, −10.00257223112120863273026569690, −8.718124379129775988561450381670, −7.32218074868377224800713432218, −6.17002023114334092214272956051, −5.53552409892545467422725231422, −4.09571466549271460036757872632, −2.77501745875736857317829583381, −0.46252998525701148991830961259,
1.07338386481954633477665586476, 2.89305351647530005229924170721, 4.65853733532574879750641609111, 5.60736768814862337148311643672, 6.76898043192221313932779071766, 7.45353318584261695991259951518, 8.872320639703134433989810703064, 10.06171234150427377253512687624, 10.66653078619890535413188660253, 11.89121775978382388967515675402
