| 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
−17.42045937616970865717127865721, −15.97206674553833838234634739474, −15.12989529957310768836474736389, −13.61515275167258328429250245216, −12.89462125190470536611445352019, −10.34023976078613781513164129918, −9.562379281006166989311836696349, −8.390174188526299258668066322640, −5.37762994344362020452195081401, −4.05280461905818911838133981185,
2.98308562519359585456042899396, 5.95465591422383137600192314672, 7.72478082334823273571311545466, 8.943864253588715291859147681135, 10.80226016782278228458881179073, 12.78031403743132149517782897079, 13.50815504744730489840614896343, 14.27009152317050877874299191371, 16.03932968680767789972605245112, 17.95564583861693710351875462704
