| L(s) = 1 | + (2.77 − 15.7i)2-s + (147. + 123. i)3-s + (−240. − 87.5i)4-s + (2.32e3 − 844. i)5-s + (2.35e3 − 1.97e3i)6-s + (2.79e3 + 4.83e3i)7-s + (−2.04e3 + 3.54e3i)8-s + (2.99e3 + 1.69e4i)9-s + (−6.86e3 − 3.89e4i)10-s + (3.44e4 − 5.97e4i)11-s + (−2.45e4 − 4.25e4i)12-s + (−1.22e5 + 1.03e5i)13-s + (8.39e4 − 3.05e4i)14-s + (4.45e5 + 1.62e5i)15-s + (5.02e4 + 4.21e4i)16-s + (−8.04e4 + 4.56e5i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (1.04 + 0.880i)3-s + (−0.469 − 0.171i)4-s + (1.66 − 0.604i)5-s + (0.741 − 0.622i)6-s + (0.439 + 0.761i)7-s + (−0.176 + 0.306i)8-s + (0.151 + 0.861i)9-s + (−0.217 − 1.23i)10-s + (0.710 − 1.23i)11-s + (−0.342 − 0.592i)12-s + (−1.19 + 1.00i)13-s + (0.584 − 0.212i)14-s + (2.27 + 0.827i)15-s + (0.191 + 0.160i)16-s + (−0.233 + 1.32i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.68841 - 0.427142i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.68841 - 0.427142i\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + (-3.82e5 + 4.20e5i)T \) |
| good | 3 | \( 1 + (-147. - 123. i)T + (3.41e3 + 1.93e4i)T^{2} \) |
| 5 | \( 1 + (-2.32e3 + 844. i)T + (1.49e6 - 1.25e6i)T^{2} \) |
| 7 | \( 1 + (-2.79e3 - 4.83e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-3.44e4 + 5.97e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + (1.22e5 - 1.03e5i)T + (1.84e9 - 1.04e10i)T^{2} \) |
| 17 | \( 1 + (8.04e4 - 4.56e5i)T + (-1.11e11 - 4.05e10i)T^{2} \) |
| 23 | \( 1 + (-5.80e5 - 2.11e5i)T + (1.37e12 + 1.15e12i)T^{2} \) |
| 29 | \( 1 + (5.19e5 + 2.94e6i)T + (-1.36e13 + 4.96e12i)T^{2} \) |
| 31 | \( 1 + (-3.36e4 - 5.82e4i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + 5.47e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-4.18e6 - 3.50e6i)T + (5.68e13 + 3.22e14i)T^{2} \) |
| 43 | \( 1 + (1.13e7 - 4.11e6i)T + (3.85e14 - 3.23e14i)T^{2} \) |
| 47 | \( 1 + (-1.33e6 - 7.56e6i)T + (-1.05e15 + 3.82e14i)T^{2} \) |
| 53 | \( 1 + (-5.23e7 - 1.90e7i)T + (2.52e15 + 2.12e15i)T^{2} \) |
| 59 | \( 1 + (2.53e7 - 1.43e8i)T + (-8.14e15 - 2.96e15i)T^{2} \) |
| 61 | \( 1 + (1.95e8 + 7.09e7i)T + (8.95e15 + 7.51e15i)T^{2} \) |
| 67 | \( 1 + (-2.04e7 - 1.15e8i)T + (-2.55e16 + 9.30e15i)T^{2} \) |
| 71 | \( 1 + (2.79e8 - 1.01e8i)T + (3.51e16 - 2.94e16i)T^{2} \) |
| 73 | \( 1 + (2.85e8 + 2.39e8i)T + (1.02e16 + 5.79e16i)T^{2} \) |
| 79 | \( 1 + (1.67e8 + 1.40e8i)T + (2.08e16 + 1.18e17i)T^{2} \) |
| 83 | \( 1 + (3.34e8 + 5.79e8i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 + (-1.67e8 + 1.40e8i)T + (6.08e16 - 3.45e17i)T^{2} \) |
| 97 | \( 1 + (-7.58e7 + 4.30e8i)T + (-7.14e17 - 2.60e17i)T^{2} \) |
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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
−14.14593576985009744386017478971, −13.35495068957553298672506562194, −11.81703435996162703465205108130, −10.21575945233798305289449937330, −9.124236343111357731732655127113, −8.829451702854685859314513806311, −5.87016293870458350862399407335, −4.52117826267804248202695544816, −2.77234824804921762235092449339, −1.63384232257218964493060702002,
1.50570593750881303050341818508, 2.78692111328040662083306389935, 5.14482051263665394272682708945, 6.93367185752084433947155619525, 7.47963648254009896319837319055, 9.252848674771359597785162530050, 10.17561354571408996792126464030, 12.51250110493147673668304038633, 13.61822965646138536701245105092, 14.25544105304178151087069173578
