| L(s) = 1 | + (−2.77 − 15.7i)2-s + (41.1 − 34.5i)3-s + (−240. + 87.5i)4-s + (−479. − 174. i)5-s + (−658. − 552. i)6-s + (5.18e3 − 8.98e3i)7-s + (2.04e3 + 3.54e3i)8-s + (−2.91e3 + 1.65e4i)9-s + (−1.41e3 + 8.04e3i)10-s + (−2.10e4 − 3.64e4i)11-s + (−6.87e3 + 1.19e4i)12-s + (−7.49e4 − 6.28e4i)13-s + (−1.56e5 − 5.67e4i)14-s + (−2.57e4 + 9.38e3i)15-s + (5.02e4 − 4.21e4i)16-s + (−1.35e4 − 7.67e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (0.293 − 0.246i)3-s + (−0.469 + 0.171i)4-s + (−0.343 − 0.124i)5-s + (−0.207 − 0.174i)6-s + (0.816 − 1.41i)7-s + (0.176 + 0.306i)8-s + (−0.148 + 0.840i)9-s + (−0.0448 + 0.254i)10-s + (−0.433 − 0.751i)11-s + (−0.0957 + 0.165i)12-s + (−0.727 − 0.610i)13-s + (−1.08 − 0.395i)14-s + (−0.131 + 0.0478i)15-s + (0.191 − 0.160i)16-s + (−0.0393 − 0.222i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.838 - 0.544i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.204819 + 0.692152i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.204819 + 0.692152i\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + (3.83e4 - 5.66e5i)T \) |
| good | 3 | \( 1 + (-41.1 + 34.5i)T + (3.41e3 - 1.93e4i)T^{2} \) |
| 5 | \( 1 + (479. + 174. i)T + (1.49e6 + 1.25e6i)T^{2} \) |
| 7 | \( 1 + (-5.18e3 + 8.98e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (2.10e4 + 3.64e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (7.49e4 + 6.28e4i)T + (1.84e9 + 1.04e10i)T^{2} \) |
| 17 | \( 1 + (1.35e4 + 7.67e4i)T + (-1.11e11 + 4.05e10i)T^{2} \) |
| 23 | \( 1 + (1.48e6 - 5.41e5i)T + (1.37e12 - 1.15e12i)T^{2} \) |
| 29 | \( 1 + (-1.02e5 + 5.79e5i)T + (-1.36e13 - 4.96e12i)T^{2} \) |
| 31 | \( 1 + (4.61e6 - 7.99e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + 1.15e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-1.71e6 + 1.43e6i)T + (5.68e13 - 3.22e14i)T^{2} \) |
| 43 | \( 1 + (-1.75e7 - 6.38e6i)T + (3.85e14 + 3.23e14i)T^{2} \) |
| 47 | \( 1 + (-8.76e6 + 4.96e7i)T + (-1.05e15 - 3.82e14i)T^{2} \) |
| 53 | \( 1 + (2.71e7 - 9.88e6i)T + (2.52e15 - 2.12e15i)T^{2} \) |
| 59 | \( 1 + (2.52e7 + 1.42e8i)T + (-8.14e15 + 2.96e15i)T^{2} \) |
| 61 | \( 1 + (-5.06e7 + 1.84e7i)T + (8.95e15 - 7.51e15i)T^{2} \) |
| 67 | \( 1 + (-2.94e7 + 1.66e8i)T + (-2.55e16 - 9.30e15i)T^{2} \) |
| 71 | \( 1 + (2.13e8 + 7.77e7i)T + (3.51e16 + 2.94e16i)T^{2} \) |
| 73 | \( 1 + (-3.38e8 + 2.84e8i)T + (1.02e16 - 5.79e16i)T^{2} \) |
| 79 | \( 1 + (-1.93e8 + 1.62e8i)T + (2.08e16 - 1.18e17i)T^{2} \) |
| 83 | \( 1 + (4.15e8 - 7.18e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (1.43e8 + 1.20e8i)T + (6.08e16 + 3.45e17i)T^{2} \) |
| 97 | \( 1 + (4.86e7 + 2.75e8i)T + (-7.14e17 + 2.60e17i)T^{2} \) |
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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
−13.71639362427410357208493501170, −12.32989296353383211861711518335, −10.96170193453370362582481067824, −10.22068118151225119398794153194, −8.227285848257308109152129023346, −7.59449659990813144400788365971, −5.09374281107318705425533478464, −3.62239919849092292156542092690, −1.82175971656848578374127645207, −0.25356571304868097459632413183,
2.27929244639051772508773869355, 4.37278848265977174585439003244, 5.78530034933342835329275077198, 7.41274712531642273202208595311, 8.704302608491531006438069156284, 9.625187564411382659190841934592, 11.52358655062878935177715574200, 12.53897503679186650842547953488, 14.34459856352450414054815568516, 15.11825764534914650216866951852
