| L(s) = 1 | + (2.77 − 15.7i)2-s + (−127. − 106. i)3-s + (−240. − 87.5i)4-s + (−1.97e3 + 717. i)5-s + (−2.03e3 + 1.70e3i)6-s + (−5.18e3 − 8.98e3i)7-s + (−2.04e3 + 3.54e3i)8-s + (1.37e3 + 7.80e3i)9-s + (5.83e3 + 3.30e4i)10-s + (−6.50e3 + 1.12e4i)11-s + (2.12e4 + 3.68e4i)12-s + (−6.56e4 + 5.50e4i)13-s + (−1.55e5 + 5.67e4i)14-s + (3.27e5 + 1.19e5i)15-s + (5.02e4 + 4.21e4i)16-s + (7.73e4 − 4.38e5i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (−0.907 − 0.761i)3-s + (−0.469 − 0.171i)4-s + (−1.41 + 0.513i)5-s + (−0.641 + 0.538i)6-s + (−0.816 − 1.41i)7-s + (−0.176 + 0.306i)8-s + (0.0699 + 0.396i)9-s + (0.184 + 1.04i)10-s + (−0.134 + 0.232i)11-s + (0.296 + 0.512i)12-s + (−0.637 + 0.535i)13-s + (−1.08 + 0.394i)14-s + (1.67 + 0.608i)15-s + (0.191 + 0.160i)16-s + (0.224 − 1.27i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.0364960 + 0.00480245i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0364960 + 0.00480245i\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + (-1.38e5 + 5.50e5i)T \) |
| good | 3 | \( 1 + (127. + 106. i)T + (3.41e3 + 1.93e4i)T^{2} \) |
| 5 | \( 1 + (1.97e3 - 717. i)T + (1.49e6 - 1.25e6i)T^{2} \) |
| 7 | \( 1 + (5.18e3 + 8.98e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (6.50e3 - 1.12e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + (6.56e4 - 5.50e4i)T + (1.84e9 - 1.04e10i)T^{2} \) |
| 17 | \( 1 + (-7.73e4 + 4.38e5i)T + (-1.11e11 - 4.05e10i)T^{2} \) |
| 23 | \( 1 + (-7.29e5 - 2.65e5i)T + (1.37e12 + 1.15e12i)T^{2} \) |
| 29 | \( 1 + (1.11e6 + 6.33e6i)T + (-1.36e13 + 4.96e12i)T^{2} \) |
| 31 | \( 1 + (-2.34e6 - 4.06e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + 1.59e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-2.44e7 - 2.05e7i)T + (5.68e13 + 3.22e14i)T^{2} \) |
| 43 | \( 1 + (8.63e6 - 3.14e6i)T + (3.85e14 - 3.23e14i)T^{2} \) |
| 47 | \( 1 + (2.53e6 + 1.43e7i)T + (-1.05e15 + 3.82e14i)T^{2} \) |
| 53 | \( 1 + (7.07e7 + 2.57e7i)T + (2.52e15 + 2.12e15i)T^{2} \) |
| 59 | \( 1 + (-4.80e6 + 2.72e7i)T + (-8.14e15 - 2.96e15i)T^{2} \) |
| 61 | \( 1 + (8.30e7 + 3.02e7i)T + (8.95e15 + 7.51e15i)T^{2} \) |
| 67 | \( 1 + (-2.88e7 - 1.63e8i)T + (-2.55e16 + 9.30e15i)T^{2} \) |
| 71 | \( 1 + (1.71e8 - 6.23e7i)T + (3.51e16 - 2.94e16i)T^{2} \) |
| 73 | \( 1 + (-1.26e8 - 1.05e8i)T + (1.02e16 + 5.79e16i)T^{2} \) |
| 79 | \( 1 + (-2.65e8 - 2.22e8i)T + (2.08e16 + 1.18e17i)T^{2} \) |
| 83 | \( 1 + (6.06e7 + 1.04e8i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 + (2.00e8 - 1.68e8i)T + (6.08e16 - 3.45e17i)T^{2} \) |
| 97 | \( 1 + (-1.98e8 + 1.12e9i)T + (-7.14e17 - 2.60e17i)T^{2} \) |
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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
−14.00028301057549458112015545880, −12.85249836759309977767085599735, −11.74824433616529113441127184480, −11.14698647017604196564981059031, −9.744861349721590514759312297416, −7.44849919742769573556700987793, −6.82106578040534686887232438072, −4.57850282783994668398356352848, −3.18471086809671699701547357729, −0.71755213196221085272020662193,
0.02426643258765045748819628398, 3.54023979851536856759859374689, 5.00870275591735083923878713497, 5.98335525103283105984362206151, 7.86274469895189807435357120937, 9.043848901865473834672193278334, 10.60185014359950779899494060470, 12.11755648778452424205267913976, 12.59355890602673490035940958667, 14.88016750397670732487008358363
