| L(s) = 1 | + (0.287 − 0.0455i)2-s + (1.72 − 3.39i)3-s + (−3.72 + 1.20i)4-s + (2.36 + 4.40i)5-s + (0.342 − 1.05i)6-s + (−2.38 + 2.38i)7-s + (−2.05 + 1.04i)8-s + (−3.22 − 4.43i)9-s + (0.880 + 1.15i)10-s + (−15.4 − 11.2i)11-s + (−2.33 + 14.7i)12-s + (16.2 + 2.56i)13-s + (−0.578 + 0.796i)14-s + (19.0 − 0.404i)15-s + (12.1 − 8.81i)16-s + (−2.10 − 4.12i)17-s + ⋯ |
| L(s) = 1 | + (0.143 − 0.0227i)2-s + (0.575 − 1.13i)3-s + (−0.930 + 0.302i)4-s + (0.472 + 0.881i)5-s + (0.0570 − 0.175i)6-s + (−0.341 + 0.341i)7-s + (−0.256 + 0.130i)8-s + (−0.358 − 0.493i)9-s + (0.0880 + 0.115i)10-s + (−1.40 − 1.02i)11-s + (−0.194 + 1.22i)12-s + (1.24 + 0.197i)13-s + (−0.0413 + 0.0568i)14-s + (1.26 − 0.0269i)15-s + (0.757 − 0.550i)16-s + (−0.123 − 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.977724 - 0.196838i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.977724 - 0.196838i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.36 - 4.40i)T \) |
| good | 2 | \( 1 + (-0.287 + 0.0455i)T + (3.80 - 1.23i)T^{2} \) |
| 3 | \( 1 + (-1.72 + 3.39i)T + (-5.29 - 7.28i)T^{2} \) |
| 7 | \( 1 + (2.38 - 2.38i)T - 49iT^{2} \) |
| 11 | \( 1 + (15.4 + 11.2i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (-16.2 - 2.56i)T + (160. + 52.2i)T^{2} \) |
| 17 | \( 1 + (2.10 + 4.12i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (-1.02 - 0.333i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + (1.81 + 11.4i)T + (-503. + 163. i)T^{2} \) |
| 29 | \( 1 + (17.5 - 5.70i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (6.76 - 20.8i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (-7.13 + 45.0i)T + (-1.30e3 - 423. i)T^{2} \) |
| 41 | \( 1 + (13.3 - 9.71i)T + (519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-41.9 - 41.9i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-20.1 - 10.2i)T + (1.29e3 + 1.78e3i)T^{2} \) |
| 53 | \( 1 + (21.6 - 42.4i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (27.2 + 37.5i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-45.3 - 32.9i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-2.50 - 4.92i)T + (-2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (20.2 + 62.4i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (16.1 + 101. i)T + (-5.06e3 + 1.64e3i)T^{2} \) |
| 79 | \( 1 + (81.7 - 26.5i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-19.1 + 9.77i)T + (4.04e3 - 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-52.7 + 72.5i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (83.7 + 42.6i)T + (5.53e3 + 7.61e3i)T^{2} \) |
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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
−17.95564583861693710351875462704, −16.03932968680767789972605245112, −14.27009152317050877874299191371, −13.50815504744730489840614896343, −12.78031403743132149517782897079, −10.80226016782278228458881179073, −8.943864253588715291859147681135, −7.72478082334823273571311545466, −5.95465591422383137600192314672, −2.98308562519359585456042899396,
4.05280461905818911838133981185, 5.37762994344362020452195081401, 8.390174188526299258668066322640, 9.562379281006166989311836696349, 10.34023976078613781513164129918, 12.89462125190470536611445352019, 13.61515275167258328429250245216, 15.12989529957310768836474736389, 15.97206674553833838234634739474, 17.42045937616970865717127865721
