| L(s) = 1 | + (12.2 − 10.2i)2-s + (−106. − 38.6i)3-s + (44.4 − 252. i)4-s + (264. + 1.49e3i)5-s + (−1.69e3 + 618. i)6-s + (1.56e3 − 2.70e3i)7-s + (−2.04e3 − 3.54e3i)8-s + (−5.28e3 − 4.43e3i)9-s + (1.86e4 + 1.56e4i)10-s + (4.43e4 + 7.67e4i)11-s + (−1.44e4 + 2.50e4i)12-s + (4.89e4 − 1.78e4i)13-s + (−8.67e3 − 4.92e4i)14-s + (2.98e4 − 1.69e5i)15-s + (−6.15e4 − 2.24e4i)16-s + (2.62e5 − 2.20e5i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (−0.757 − 0.275i)3-s + (0.0868 − 0.492i)4-s + (0.189 + 1.07i)5-s + (−0.535 + 0.194i)6-s + (0.245 − 0.425i)7-s + (−0.176 − 0.306i)8-s + (−0.268 − 0.225i)9-s + (0.589 + 0.494i)10-s + (0.912 + 1.58i)11-s + (−0.201 + 0.348i)12-s + (0.475 − 0.172i)13-s + (−0.0603 − 0.342i)14-s + (0.152 − 0.864i)15-s + (−0.234 − 0.0855i)16-s + (0.763 − 0.640i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 + 0.648i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.760 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.05100 - 0.755768i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.05100 - 0.755768i\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + (-4.32e5 + 3.68e5i)T \) |
| good | 3 | \( 1 + (106. + 38.6i)T + (1.50e4 + 1.26e4i)T^{2} \) |
| 5 | \( 1 + (-264. - 1.49e3i)T + (-1.83e6 + 6.68e5i)T^{2} \) |
| 7 | \( 1 + (-1.56e3 + 2.70e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-4.43e4 - 7.67e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-4.89e4 + 1.78e4i)T + (8.12e9 - 6.81e9i)T^{2} \) |
| 17 | \( 1 + (-2.62e5 + 2.20e5i)T + (2.05e10 - 1.16e11i)T^{2} \) |
| 23 | \( 1 + (-4.38e5 + 2.48e6i)T + (-1.69e12 - 6.16e11i)T^{2} \) |
| 29 | \( 1 + (8.60e5 + 7.22e5i)T + (2.51e12 + 1.42e13i)T^{2} \) |
| 31 | \( 1 + (1.38e6 - 2.39e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 - 2.09e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-5.45e6 - 1.98e6i)T + (2.50e14 + 2.10e14i)T^{2} \) |
| 43 | \( 1 + (-7.14e6 - 4.05e7i)T + (-4.72e14 + 1.71e14i)T^{2} \) |
| 47 | \( 1 + (1.78e7 + 1.49e7i)T + (1.94e14 + 1.10e15i)T^{2} \) |
| 53 | \( 1 + (-4.68e6 + 2.65e7i)T + (-3.10e15 - 1.12e15i)T^{2} \) |
| 59 | \( 1 + (5.18e7 - 4.35e7i)T + (1.50e15 - 8.53e15i)T^{2} \) |
| 61 | \( 1 + (5.61e6 - 3.18e7i)T + (-1.09e16 - 3.99e15i)T^{2} \) |
| 67 | \( 1 + (-1.57e8 - 1.31e8i)T + (4.72e15 + 2.67e16i)T^{2} \) |
| 71 | \( 1 + (6.38e7 + 3.62e8i)T + (-4.30e16 + 1.56e16i)T^{2} \) |
| 73 | \( 1 + (-7.95e6 - 2.89e6i)T + (4.50e16 + 3.78e16i)T^{2} \) |
| 79 | \( 1 + (2.05e8 + 7.47e7i)T + (9.18e16 + 7.70e16i)T^{2} \) |
| 83 | \( 1 + (1.22e8 - 2.12e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (-2.68e8 + 9.78e7i)T + (2.68e17 - 2.25e17i)T^{2} \) |
| 97 | \( 1 + (-1.01e9 + 8.51e8i)T + (1.32e17 - 7.48e17i)T^{2} \) |
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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
−14.34279217150427176194359448662, −12.75134197177024602566970450285, −11.69784240699113641437224956279, −10.81209616476888594799447078247, −9.586868962310095825950019519270, −7.17175528205809272384134583477, −6.26050083705143519685399975035, −4.59516604348475914970947170901, −2.84127949367935165106640809855, −1.01032747303669693457093734715,
1.10451738724998675745691238475, 3.70026456944976867934031829847, 5.41446097304892487499972092074, 5.89061180039054110452072401483, 8.091073212133085330939212321877, 9.204564961187773669355926610599, 11.19908878905510696422863491444, 11.95807519433361846624215419250, 13.32655978019155570821445530298, 14.32722881559535904088584391471
