L(s) = 1 | + (−8 − 13.8i)2-s + (−43.8 − 75.9i)3-s + (−127. + 221. i)4-s + (993. + 1.72e3i)5-s + (−701. + 1.21e3i)6-s − 8.24e3·7-s + 4.09e3·8-s + (5.99e3 − 1.03e4i)9-s + (1.58e4 − 2.75e4i)10-s + 6.24e4·11-s + 2.24e4·12-s + (−1.25e3 + 2.17e3i)13-s + (6.59e4 + 1.14e5i)14-s + (8.71e4 − 1.50e5i)15-s + (−3.27e4 − 5.67e4i)16-s + (−2.80e5 − 4.86e5i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.312 − 0.541i)3-s + (−0.249 + 0.433i)4-s + (0.711 + 1.23i)5-s + (−0.221 + 0.382i)6-s − 1.29·7-s + 0.353·8-s + (0.304 − 0.527i)9-s + (0.502 − 0.870i)10-s + 1.28·11-s + 0.312·12-s + (−0.0122 + 0.0211i)13-s + (0.458 + 0.794i)14-s + (0.444 − 0.769i)15-s + (−0.125 − 0.216i)16-s + (−0.815 − 1.41i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 + 0.655i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.754 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.321554 - 0.860392i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.321554 - 0.860392i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (8 + 13.8i)T \) |
| 19 | \( 1 + (3.40e5 - 4.54e5i)T \) |
good | 3 | \( 1 + (43.8 + 75.9i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-993. - 1.72e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 + 8.24e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 6.24e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (1.25e3 - 2.17e3i)T + (-5.30e9 - 9.18e9i)T^{2} \) |
| 17 | \( 1 + (2.80e5 + 4.86e5i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 23 | \( 1 + (-7.84e5 + 1.35e6i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (-2.89e6 + 5.01e6i)T + (-7.25e12 - 1.25e13i)T^{2} \) |
| 31 | \( 1 + 8.75e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.54e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-6.53e6 - 1.13e7i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 + (1.61e7 + 2.79e7i)T + (-2.51e14 + 4.35e14i)T^{2} \) |
| 47 | \( 1 + (-2.96e5 + 5.13e5i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-5.45e7 + 9.44e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (2.22e7 + 3.84e7i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-5.34e7 + 9.25e7i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-5.28e7 + 9.15e7i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (6.30e7 + 1.09e8i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 + (-1.21e8 - 2.10e8i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (1.40e8 + 2.43e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 - 6.00e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-7.34e7 + 1.27e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + (3.87e8 + 6.71e8i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
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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
−13.61291732638946899193944167513, −12.52747408636343796318388846656, −11.38863049924167647690472993513, −10.04610556810318623805094821977, −9.233787047684407220757575010596, −6.86781784596123823333308858073, −6.41671393028931453084547132972, −3.62676182434036552769729151728, −2.23222295175448727304981133062, −0.40878182037391628007356011007,
1.41595147884850994681773173690, 4.18081878707076167731095970701, 5.56752838255112986615462694533, 6.76995052027596228394573532628, 8.871782365524604707625546157124, 9.444607100030314910755146678862, 10.74500574555722124800402847004, 12.71094192386526896399259860090, 13.41043153479542799218937485496, 15.08149384508862697401886236085