| L(s) = 1 | + (−12.2 − 10.2i)2-s + (−138. + 50.3i)3-s + (44.4 + 252. i)4-s + (84.9 − 481. i)5-s + (2.21e3 + 805. i)6-s + (−3.52e3 − 6.10e3i)7-s + (2.04e3 − 3.54e3i)8-s + (1.52e3 − 1.27e3i)9-s + (−5.99e3 + 5.02e3i)10-s + (3.49e4 − 6.05e4i)11-s + (−1.88e4 − 3.26e4i)12-s + (−1.74e5 − 6.36e4i)13-s + (−1.95e4 + 1.11e5i)14-s + (1.24e4 + 7.08e4i)15-s + (−6.15e4 + 2.24e4i)16-s + (3.77e5 + 3.16e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (−0.985 + 0.358i)3-s + (0.0868 + 0.492i)4-s + (0.0607 − 0.344i)5-s + (0.697 + 0.253i)6-s + (−0.555 − 0.961i)7-s + (0.176 − 0.306i)8-s + (0.0773 − 0.0648i)9-s + (−0.189 + 0.159i)10-s + (0.720 − 1.24i)11-s + (−0.262 − 0.454i)12-s + (−1.69 − 0.618i)13-s + (−0.136 + 0.773i)14-s + (0.0637 + 0.361i)15-s + (−0.234 + 0.0855i)16-s + (1.09 + 0.919i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.259564 + 0.183918i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.259564 + 0.183918i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (12.2 + 10.2i)T \) |
| 19 | \( 1 + (5.59e5 + 1.00e5i)T \) |
| good | 3 | \( 1 + (138. - 50.3i)T + (1.50e4 - 1.26e4i)T^{2} \) |
| 5 | \( 1 + (-84.9 + 481. i)T + (-1.83e6 - 6.68e5i)T^{2} \) |
| 7 | \( 1 + (3.52e3 + 6.10e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-3.49e4 + 6.05e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + (1.74e5 + 6.36e4i)T + (8.12e9 + 6.81e9i)T^{2} \) |
| 17 | \( 1 + (-3.77e5 - 3.16e5i)T + (2.05e10 + 1.16e11i)T^{2} \) |
| 23 | \( 1 + (-1.18e5 - 6.74e5i)T + (-1.69e12 + 6.16e11i)T^{2} \) |
| 29 | \( 1 + (1.48e6 - 1.24e6i)T + (2.51e12 - 1.42e13i)T^{2} \) |
| 31 | \( 1 + (-4.33e6 - 7.50e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + 1.16e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (1.80e7 - 6.56e6i)T + (2.50e14 - 2.10e14i)T^{2} \) |
| 43 | \( 1 + (5.70e6 - 3.23e7i)T + (-4.72e14 - 1.71e14i)T^{2} \) |
| 47 | \( 1 + (6.85e6 - 5.74e6i)T + (1.94e14 - 1.10e15i)T^{2} \) |
| 53 | \( 1 + (1.89e7 + 1.07e8i)T + (-3.10e15 + 1.12e15i)T^{2} \) |
| 59 | \( 1 + (-8.63e7 - 7.24e7i)T + (1.50e15 + 8.53e15i)T^{2} \) |
| 61 | \( 1 + (-2.25e6 - 1.28e7i)T + (-1.09e16 + 3.99e15i)T^{2} \) |
| 67 | \( 1 + (-1.00e8 + 8.46e7i)T + (4.72e15 - 2.67e16i)T^{2} \) |
| 71 | \( 1 + (5.80e7 - 3.29e8i)T + (-4.30e16 - 1.56e16i)T^{2} \) |
| 73 | \( 1 + (6.92e7 - 2.51e7i)T + (4.50e16 - 3.78e16i)T^{2} \) |
| 79 | \( 1 + (-5.10e8 + 1.85e8i)T + (9.18e16 - 7.70e16i)T^{2} \) |
| 83 | \( 1 + (-1.94e8 - 3.36e8i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 + (1.35e8 + 4.93e7i)T + (2.68e17 + 2.25e17i)T^{2} \) |
| 97 | \( 1 + (-6.99e8 - 5.87e8i)T + (1.32e17 + 7.48e17i)T^{2} \) |
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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
−14.51412728095252642197734539533, −12.97741354315220994857099002595, −11.91396647166622388624640178806, −10.68907218716967909510963962359, −9.965889104926865838629983274305, −8.345481275157566957060779710541, −6.68367643440933110693550551082, −5.10203844952366763519686813191, −3.38278010396177589597202762933, −0.955747216842257602606269507334,
0.20171921829457488952166791205, 2.25641532457369581497422908340, 5.00262770865177799670836752910, 6.37377938347558257301896754204, 7.21742321425518515512626576365, 9.173077170751268736873737611524, 10.14683584661956204536749034161, 11.94037140881201074330331746026, 12.30160433863751906650578762217, 14.45525682847396970668123044926
