| L(s) = 1 | + (−3.06 − 2.57i)2-s + (0.577 − 0.210i)3-s + (2.77 + 15.7i)4-s + (2.13 − 12.1i)5-s + (−2.30 − 0.840i)6-s + (−76.5 − 132. i)7-s + (32.0 − 55.4i)8-s + (−185. + 155. i)9-s + (−37.6 + 31.5i)10-s + (−240. + 416. i)11-s + (4.91 + 8.50i)12-s + (−644. − 234. i)13-s + (−106. + 603. i)14-s + (−1.31 − 7.43i)15-s + (−240. + 87.5i)16-s + (−9.99e2 − 838. i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (0.0370 − 0.0134i)3-s + (0.0868 + 0.492i)4-s + (0.0381 − 0.216i)5-s + (−0.0261 − 0.00952i)6-s + (−0.590 − 1.02i)7-s + (0.176 − 0.306i)8-s + (−0.764 + 0.641i)9-s + (−0.119 + 0.0999i)10-s + (−0.598 + 1.03i)11-s + (0.00984 + 0.0170i)12-s + (−1.05 − 0.384i)13-s + (−0.145 + 0.822i)14-s + (−0.00150 − 0.00852i)15-s + (−0.234 + 0.0855i)16-s + (−0.838 − 0.703i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.257i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0244104 + 0.186691i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0244104 + 0.186691i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.06 + 2.57i)T \) |
| 19 | \( 1 + (158. - 1.56e3i)T \) |
| good | 3 | \( 1 + (-0.577 + 0.210i)T + (186. - 156. i)T^{2} \) |
| 5 | \( 1 + (-2.13 + 12.1i)T + (-2.93e3 - 1.06e3i)T^{2} \) |
| 7 | \( 1 + (76.5 + 132. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (240. - 416. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (644. + 234. i)T + (2.84e5 + 2.38e5i)T^{2} \) |
| 17 | \( 1 + (9.99e2 + 838. i)T + (2.46e5 + 1.39e6i)T^{2} \) |
| 23 | \( 1 + (754. + 4.27e3i)T + (-6.04e6 + 2.20e6i)T^{2} \) |
| 29 | \( 1 + (-3.61e3 + 3.03e3i)T + (3.56e6 - 2.01e7i)T^{2} \) |
| 31 | \( 1 + (-225. - 389. i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + 5.23e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-253. + 92.2i)T + (8.87e7 - 7.44e7i)T^{2} \) |
| 43 | \( 1 + (1.63e3 - 9.25e3i)T + (-1.38e8 - 5.02e7i)T^{2} \) |
| 47 | \( 1 + (2.06e3 - 1.73e3i)T + (3.98e7 - 2.25e8i)T^{2} \) |
| 53 | \( 1 + (4.77e3 + 2.70e4i)T + (-3.92e8 + 1.43e8i)T^{2} \) |
| 59 | \( 1 + (-2.13e4 - 1.78e4i)T + (1.24e8 + 7.04e8i)T^{2} \) |
| 61 | \( 1 + (2.37e3 + 1.34e4i)T + (-7.93e8 + 2.88e8i)T^{2} \) |
| 67 | \( 1 + (5.24e4 - 4.39e4i)T + (2.34e8 - 1.32e9i)T^{2} \) |
| 71 | \( 1 + (-2.29e3 + 1.30e4i)T + (-1.69e9 - 6.17e8i)T^{2} \) |
| 73 | \( 1 + (-7.68e4 + 2.79e4i)T + (1.58e9 - 1.33e9i)T^{2} \) |
| 79 | \( 1 + (3.21e4 - 1.17e4i)T + (2.35e9 - 1.97e9i)T^{2} \) |
| 83 | \( 1 + (2.13e4 + 3.69e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + (5.36e4 + 1.95e4i)T + (4.27e9 + 3.58e9i)T^{2} \) |
| 97 | \( 1 + (5.01e4 + 4.20e4i)T + (1.49e9 + 8.45e9i)T^{2} \) |
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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
−14.56113717541529512540543393208, −13.24381663599126115684073937897, −12.23218595410788876792177278187, −10.63957855385721087032693662671, −9.869302440233277400273671674522, −8.238541288067807461013193165150, −6.98923741222788668537122686583, −4.69064980145113284596614454325, −2.55499354444686674176368838718, −0.11759454141334719616698984200,
2.79579832922109523893367180289, 5.51128951926801601936816653710, 6.72455601486398021292688907276, 8.524404260023168047751692542745, 9.374777668309350975363406338203, 10.95197496302787592609655381877, 12.22173253751176513815241468554, 13.75851241015641527920232675950, 15.06234069496897521982828561641, 15.81268004839002811628624591490
