L(s) = 1 | + (3.34 + 5.79i)5-s + (12.7 − 13.4i)7-s + (28.2 + 16.3i)11-s − 67.9i·13-s + (15.3 − 26.5i)17-s + (21.8 − 12.6i)19-s + (−68.6 + 39.6i)23-s + (40.0 − 69.4i)25-s − 109. i·29-s + (−238. − 137. i)31-s + (120. + 28.8i)35-s + (160. + 277. i)37-s + 184.·41-s − 364.·43-s + (25.7 + 44.6i)47-s + ⋯ |
L(s) = 1 | + (0.299 + 0.518i)5-s + (0.687 − 0.725i)7-s + (0.775 + 0.447i)11-s − 1.44i·13-s + (0.218 − 0.378i)17-s + (0.264 − 0.152i)19-s + (−0.622 + 0.359i)23-s + (0.320 − 0.555i)25-s − 0.702i·29-s + (−1.38 − 0.797i)31-s + (0.582 + 0.139i)35-s + (0.711 + 1.23i)37-s + 0.704·41-s − 1.29·43-s + (0.0799 + 0.138i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.317 + 0.948i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.244572236\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.244572236\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-12.7 + 13.4i)T \) |
good | 5 | \( 1 + (-3.34 - 5.79i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-28.2 - 16.3i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 67.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-15.3 + 26.5i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-21.8 + 12.6i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (68.6 - 39.6i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 109. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (238. + 137. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-160. - 277. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 184.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 364.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-25.7 - 44.6i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (532. + 307. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-207. + 359. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-411. + 237. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (142. - 246. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 965. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-225. - 130. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (219. + 379. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 76.4T + 5.71e5T^{2} \) |
| 89 | \( 1 + (356. + 617. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 410. iT - 9.12e5T^{2} \) |
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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
−9.821345694334335954784577173476, −8.409402770906469510177557755263, −7.73552779770868778774819805945, −6.96577523541170352777287962940, −6.01771995935067168303711648103, −5.06131450815995078466249548349, −4.05991286503265833697098078870, −3.04398646689761324720519784038, −1.79703937713067332627742128648, −0.57649019606367127732004504222,
1.30130846127997634456185086567, 2.04487409787690014097696573292, 3.55128317958334075575429356174, 4.54588919714721272872681917022, 5.44462727129573306052884171592, 6.24291297280748503440203040405, 7.21914508113508554469975792179, 8.272628376763459643850970159233, 9.121490583147923053941677052735, 9.308306179420052254682602122279