| L(s) = 1 | + (0.967 − 2.65i)2-s + (3.41 − 0.602i)3-s + (−6.12 − 5.14i)4-s + (10.5 − 8.86i)5-s + (1.70 − 9.66i)6-s + (40.2 − 69.7i)7-s + (−19.5 + 11.3i)8-s + (−64.7 + 23.5i)9-s + (−13.3 − 36.6i)10-s + (−1.61 − 2.79i)11-s + (−24.0 − 13.8i)12-s + (122. + 21.6i)13-s + (−146. − 174. i)14-s + (30.7 − 36.6i)15-s + (11.1 + 63.0i)16-s + (108. + 39.4i)17-s + ⋯ |
| L(s) = 1 | + (0.241 − 0.664i)2-s + (0.379 − 0.0669i)3-s + (−0.383 − 0.321i)4-s + (0.422 − 0.354i)5-s + (0.0473 − 0.268i)6-s + (0.822 − 1.42i)7-s + (−0.306 + 0.176i)8-s + (−0.799 + 0.291i)9-s + (−0.133 − 0.366i)10-s + (−0.0133 − 0.0230i)11-s + (−0.166 − 0.0964i)12-s + (0.726 + 0.128i)13-s + (−0.747 − 0.890i)14-s + (0.136 − 0.163i)15-s + (0.0434 + 0.246i)16-s + (0.374 + 0.136i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0696 + 0.997i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0696 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.36690 - 1.27483i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.36690 - 1.27483i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.967 + 2.65i)T \) |
| 19 | \( 1 + (118. - 341. i)T \) |
| good | 3 | \( 1 + (-3.41 + 0.602i)T + (76.1 - 27.7i)T^{2} \) |
| 5 | \( 1 + (-10.5 + 8.86i)T + (108. - 615. i)T^{2} \) |
| 7 | \( 1 + (-40.2 + 69.7i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (1.61 + 2.79i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-122. - 21.6i)T + (2.68e4 + 9.76e3i)T^{2} \) |
| 17 | \( 1 + (-108. - 39.4i)T + (6.39e4 + 5.36e4i)T^{2} \) |
| 23 | \( 1 + (-773. - 648. i)T + (4.85e4 + 2.75e5i)T^{2} \) |
| 29 | \( 1 + (53.8 + 147. i)T + (-5.41e5 + 4.54e5i)T^{2} \) |
| 31 | \( 1 + (547. + 316. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 1.48e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (1.33e3 - 236. i)T + (2.65e6 - 9.66e5i)T^{2} \) |
| 43 | \( 1 + (96.9 - 81.3i)T + (5.93e5 - 3.36e6i)T^{2} \) |
| 47 | \( 1 + (-1.99e3 + 727. i)T + (3.73e6 - 3.13e6i)T^{2} \) |
| 53 | \( 1 + (-605. + 721. i)T + (-1.37e6 - 7.77e6i)T^{2} \) |
| 59 | \( 1 + (1.66e3 - 4.57e3i)T + (-9.28e6 - 7.78e6i)T^{2} \) |
| 61 | \( 1 + (3.14e3 + 2.63e3i)T + (2.40e6 + 1.36e7i)T^{2} \) |
| 67 | \( 1 + (-1.82e3 - 5.01e3i)T + (-1.54e7 + 1.29e7i)T^{2} \) |
| 71 | \( 1 + (-3.38e3 - 4.03e3i)T + (-4.41e6 + 2.50e7i)T^{2} \) |
| 73 | \( 1 + (1.05e3 + 5.98e3i)T + (-2.66e7 + 9.71e6i)T^{2} \) |
| 79 | \( 1 + (3.23e3 - 569. i)T + (3.66e7 - 1.33e7i)T^{2} \) |
| 83 | \( 1 + (-5.14e3 + 8.90e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (795. + 140. i)T + (5.89e7 + 2.14e7i)T^{2} \) |
| 97 | \( 1 + (-2.08e3 + 5.74e3i)T + (-6.78e7 - 5.69e7i)T^{2} \) |
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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
−14.81170995208966613133033419952, −13.85158300138213301803855381645, −13.17637704579507679315314986146, −11.43285449979313557972139996204, −10.55352378029733601896144943381, −9.028084050148931477232817967699, −7.65363642081251431816286227009, −5.43594401188245557233757277955, −3.72248940051224082341609008278, −1.44654460804927548625295473402,
2.75629304450125736257673873247, 5.15004399950642929339453569643, 6.41124232795759794321139198251, 8.351886705192776941061313860872, 9.066453799410511729183427074685, 11.05956647274063110582630971889, 12.36393202539599952279423792663, 13.82945859398477440033857846192, 14.79507580147394781262994628518, 15.42878215499039428924850244346
