| L(s) = 1 | + (−2.77 + 15.7i)2-s + (199. + 167. i)3-s + (−240. − 87.5i)4-s + (1.05e3 − 383. i)5-s + (−3.18e3 + 2.67e3i)6-s + (−1.72e3 − 2.98e3i)7-s + (2.04e3 − 3.54e3i)8-s + (8.34e3 + 4.73e4i)9-s + (3.11e3 + 1.76e4i)10-s + (−4.32e4 + 7.49e4i)11-s + (−3.33e4 − 5.76e4i)12-s + (8.21e3 − 6.89e3i)13-s + (5.18e4 − 1.88e4i)14-s + (2.74e5 + 9.98e4i)15-s + (5.02e4 + 4.21e4i)16-s + (−7.48e4 + 4.24e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (1.42 + 1.19i)3-s + (−0.469 − 0.171i)4-s + (0.754 − 0.274i)5-s + (−1.00 + 0.843i)6-s + (−0.271 − 0.470i)7-s + (0.176 − 0.306i)8-s + (0.423 + 2.40i)9-s + (0.0985 + 0.558i)10-s + (−0.890 + 1.54i)11-s + (−0.463 − 0.803i)12-s + (0.0797 − 0.0669i)13-s + (0.360 − 0.131i)14-s + (1.39 + 0.509i)15-s + (0.191 + 0.160i)16-s + (−0.217 + 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 - 0.527i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.755643 + 2.65046i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.755643 + 2.65046i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.77 - 15.7i)T \) |
| 19 | \( 1 + (-4.96e5 + 2.76e5i)T \) |
| good | 3 | \( 1 + (-199. - 167. i)T + (3.41e3 + 1.93e4i)T^{2} \) |
| 5 | \( 1 + (-1.05e3 + 383. i)T + (1.49e6 - 1.25e6i)T^{2} \) |
| 7 | \( 1 + (1.72e3 + 2.98e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (4.32e4 - 7.49e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-8.21e3 + 6.89e3i)T + (1.84e9 - 1.04e10i)T^{2} \) |
| 17 | \( 1 + (7.48e4 - 4.24e5i)T + (-1.11e11 - 4.05e10i)T^{2} \) |
| 23 | \( 1 + (1.03e5 + 3.76e4i)T + (1.37e12 + 1.15e12i)T^{2} \) |
| 29 | \( 1 + (6.24e5 + 3.54e6i)T + (-1.36e13 + 4.96e12i)T^{2} \) |
| 31 | \( 1 + (-2.39e6 - 4.15e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 - 1.01e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (1.31e7 + 1.10e7i)T + (5.68e13 + 3.22e14i)T^{2} \) |
| 43 | \( 1 + (-1.98e7 + 7.21e6i)T + (3.85e14 - 3.23e14i)T^{2} \) |
| 47 | \( 1 + (9.59e6 + 5.44e7i)T + (-1.05e15 + 3.82e14i)T^{2} \) |
| 53 | \( 1 + (-6.23e7 - 2.26e7i)T + (2.52e15 + 2.12e15i)T^{2} \) |
| 59 | \( 1 + (-7.14e6 + 4.05e7i)T + (-8.14e15 - 2.96e15i)T^{2} \) |
| 61 | \( 1 + (-1.69e8 - 6.18e7i)T + (8.95e15 + 7.51e15i)T^{2} \) |
| 67 | \( 1 + (2.29e7 + 1.29e8i)T + (-2.55e16 + 9.30e15i)T^{2} \) |
| 71 | \( 1 + (1.79e8 - 6.53e7i)T + (3.51e16 - 2.94e16i)T^{2} \) |
| 73 | \( 1 + (-1.02e8 - 8.59e7i)T + (1.02e16 + 5.79e16i)T^{2} \) |
| 79 | \( 1 + (1.88e8 + 1.58e8i)T + (2.08e16 + 1.18e17i)T^{2} \) |
| 83 | \( 1 + (-3.05e8 - 5.29e8i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 + (4.93e8 - 4.13e8i)T + (6.08e16 - 3.45e17i)T^{2} \) |
| 97 | \( 1 + (-2.00e7 + 1.13e8i)T + (-7.14e17 - 2.60e17i)T^{2} \) |
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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
−15.05743644351283511766251010865, −13.80869130737622010559578542100, −13.07956134043154898567130175516, −10.24275012637073699597775273142, −9.801719393398946714835544350865, −8.595735037319800300716452111537, −7.37651774591738669162904354227, −5.20633357853104704086383226857, −3.97607072890720814312614944396, −2.18710595676609779047677811602,
0.899329629124987087387680551447, 2.44300749281307209061763040140, 3.12837068364002047080068150993, 5.97495116063781613479853690357, 7.65170885541647358161329512409, 8.769783423326503681347357568154, 9.766537260035778199196655846770, 11.56122964828339587836445080979, 12.94977020641724447933377497461, 13.65945817695517864391080496830
