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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.564508461\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.564508461\) |
\(L(\frac{5}{2})\) |
|
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
−10.40463027497567232217690797027, −9.740877869908062319179838815976, −8.676705854091349691754905962220, −8.124228700984916440506368670452, −6.81122969450468860947435496042, −5.95304086450872112993551485224, −4.91739195384423799984246142862, −3.74406925621466086422400461978, −2.57984436195754145996936106048, −0.939692578522420840924574608221,
0.64700257620242349491189483054, 2.36608479775472213905830294554, 3.74189031449812352863697185130, 4.37239785047528977586597488580, 6.08548741948031402476878554534, 6.75429112234377824517644290135, 7.52094607241001194267519706714, 8.793375057189476438995977955704, 9.546193454981735346504452296717, 10.44558110013849284949411139337