| L(s) = 1 | + (−2.78 + 0.491i)2-s + (8.78 + 10.4i)3-s + (7.51 − 2.73i)4-s + (4.36 + 1.58i)5-s + (−29.6 − 24.8i)6-s + (−11.1 + 19.3i)7-s + (−19.5 + 11.3i)8-s + (−18.3 + 104. i)9-s + (−12.9 − 2.28i)10-s + (64.7 + 112. i)11-s + (94.6 + 54.6i)12-s + (−71.1 + 84.7i)13-s + (21.6 − 59.5i)14-s + (21.7 + 59.6i)15-s + (49.0 − 41.1i)16-s + (−20.4 − 116. i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 + 0.122i)2-s + (0.975 + 1.16i)3-s + (0.469 − 0.171i)4-s + (0.174 + 0.0635i)5-s + (−0.822 − 0.690i)6-s + (−0.228 + 0.395i)7-s + (−0.306 + 0.176i)8-s + (−0.226 + 1.28i)9-s + (−0.129 − 0.0228i)10-s + (0.535 + 0.927i)11-s + (0.657 + 0.379i)12-s + (−0.420 + 0.501i)13-s + (0.110 − 0.303i)14-s + (0.0965 + 0.265i)15-s + (0.191 − 0.160i)16-s + (−0.0707 − 0.401i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0237 - 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0237 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.983136 + 1.00678i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.983136 + 1.00678i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.78 - 0.491i)T \) |
| 19 | \( 1 + (0.217 + 360. i)T \) |
| good | 3 | \( 1 + (-8.78 - 10.4i)T + (-14.0 + 79.7i)T^{2} \) |
| 5 | \( 1 + (-4.36 - 1.58i)T + (478. + 401. i)T^{2} \) |
| 7 | \( 1 + (11.1 - 19.3i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-64.7 - 112. i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (71.1 - 84.7i)T + (-4.95e3 - 2.81e4i)T^{2} \) |
| 17 | \( 1 + (20.4 + 116. i)T + (-7.84e4 + 2.85e4i)T^{2} \) |
| 23 | \( 1 + (-660. + 240. i)T + (2.14e5 - 1.79e5i)T^{2} \) |
| 29 | \( 1 + (-901. - 159. i)T + (6.64e5 + 2.41e5i)T^{2} \) |
| 31 | \( 1 + (-1.07e3 - 618. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 1.92e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-714. - 851. i)T + (-4.90e5 + 2.78e6i)T^{2} \) |
| 43 | \( 1 + (3.20e3 + 1.16e3i)T + (2.61e6 + 2.19e6i)T^{2} \) |
| 47 | \( 1 + (-646. + 3.66e3i)T + (-4.58e6 - 1.66e6i)T^{2} \) |
| 53 | \( 1 + (-1.04e3 - 2.88e3i)T + (-6.04e6 + 5.07e6i)T^{2} \) |
| 59 | \( 1 + (1.03e3 - 182. i)T + (1.13e7 - 4.14e6i)T^{2} \) |
| 61 | \( 1 + (1.38e3 - 503. i)T + (1.06e7 - 8.89e6i)T^{2} \) |
| 67 | \( 1 + (5.58e3 + 985. i)T + (1.89e7 + 6.89e6i)T^{2} \) |
| 71 | \( 1 + (-1.00e3 + 2.76e3i)T + (-1.94e7 - 1.63e7i)T^{2} \) |
| 73 | \( 1 + (1.62e3 - 1.36e3i)T + (4.93e6 - 2.79e7i)T^{2} \) |
| 79 | \( 1 + (651. + 776. i)T + (-6.76e6 + 3.83e7i)T^{2} \) |
| 83 | \( 1 + (5.18e3 - 8.97e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-1.11e3 + 1.33e3i)T + (-1.08e7 - 6.17e7i)T^{2} \) |
| 97 | \( 1 + (5.87e3 - 1.03e3i)T + (8.31e7 - 3.02e7i)T^{2} \) |
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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
−15.67166244203074072489087785142, −15.00863478327840444898033775127, −13.90742711721704487723824977611, −12.03677112521638197315574188889, −10.41044186112880359589847579936, −9.442185792733954459413155696958, −8.708227758680252780876813566818, −6.92775822746912198958973115165, −4.62833097002777105444777209037, −2.64069551497080782882392766958,
1.24215254518133121333108515747, 3.10592346541519591244658079628, 6.38366564394884310471222554895, 7.70967813948720531107599710803, 8.638322693423760791026489332740, 10.02144413392387351719666962082, 11.69329206304915970244297968262, 13.00696097690218171105554337635, 13.87542441612351224868924655692, 15.12155054685891920960189903014
