| L(s) = 1 | + (12.2 + 10.2i)2-s + (242. − 88.2i)3-s + (44.4 + 252. i)4-s + (132. − 750. i)5-s + (3.88e3 + 1.41e3i)6-s + (−5.61e3 − 9.72e3i)7-s + (−2.04e3 + 3.54e3i)8-s + (3.59e4 − 3.01e4i)9-s + (9.33e3 − 7.83e3i)10-s + (8.60e3 − 1.48e4i)11-s + (3.30e4 + 5.72e4i)12-s + (−3.83e3 − 1.39e3i)13-s + (3.11e4 − 1.76e5i)14-s + (−3.41e4 − 1.93e5i)15-s + (−6.15e4 + 2.24e4i)16-s + (−4.74e4 − 3.98e4i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (1.72 − 0.629i)3-s + (0.0868 + 0.492i)4-s + (0.0946 − 0.536i)5-s + (1.22 + 0.444i)6-s + (−0.883 − 1.53i)7-s + (−0.176 + 0.306i)8-s + (1.82 − 1.53i)9-s + (0.295 − 0.247i)10-s + (0.177 − 0.306i)11-s + (0.459 + 0.796i)12-s + (−0.0372 − 0.0135i)13-s + (0.217 − 1.23i)14-s + (−0.174 − 0.987i)15-s + (−0.234 + 0.0855i)16-s + (−0.137 − 0.115i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.747i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.664 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.85441 - 1.73174i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.85441 - 1.73174i\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + (3.59e5 - 4.39e5i)T \) |
| good | 3 | \( 1 + (-242. + 88.2i)T + (1.50e4 - 1.26e4i)T^{2} \) |
| 5 | \( 1 + (-132. + 750. i)T + (-1.83e6 - 6.68e5i)T^{2} \) |
| 7 | \( 1 + (5.61e3 + 9.72e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-8.60e3 + 1.48e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + (3.83e3 + 1.39e3i)T + (8.12e9 + 6.81e9i)T^{2} \) |
| 17 | \( 1 + (4.74e4 + 3.98e4i)T + (2.05e10 + 1.16e11i)T^{2} \) |
| 23 | \( 1 + (-3.48e5 - 1.97e6i)T + (-1.69e12 + 6.16e11i)T^{2} \) |
| 29 | \( 1 + (-4.01e6 + 3.37e6i)T + (2.51e12 - 1.42e13i)T^{2} \) |
| 31 | \( 1 + (-3.78e6 - 6.55e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 - 1.22e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (1.98e7 - 7.21e6i)T + (2.50e14 - 2.10e14i)T^{2} \) |
| 43 | \( 1 + (-1.91e6 + 1.08e7i)T + (-4.72e14 - 1.71e14i)T^{2} \) |
| 47 | \( 1 + (-3.74e7 + 3.14e7i)T + (1.94e14 - 1.10e15i)T^{2} \) |
| 53 | \( 1 + (-7.32e6 - 4.15e7i)T + (-3.10e15 + 1.12e15i)T^{2} \) |
| 59 | \( 1 + (-7.86e5 - 6.60e5i)T + (1.50e15 + 8.53e15i)T^{2} \) |
| 61 | \( 1 + (1.07e7 + 6.11e7i)T + (-1.09e16 + 3.99e15i)T^{2} \) |
| 67 | \( 1 + (2.24e8 - 1.88e8i)T + (4.72e15 - 2.67e16i)T^{2} \) |
| 71 | \( 1 + (-2.27e7 + 1.29e8i)T + (-4.30e16 - 1.56e16i)T^{2} \) |
| 73 | \( 1 + (2.10e8 - 7.65e7i)T + (4.50e16 - 3.78e16i)T^{2} \) |
| 79 | \( 1 + (-2.43e8 + 8.86e7i)T + (9.18e16 - 7.70e16i)T^{2} \) |
| 83 | \( 1 + (-3.75e7 - 6.50e7i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 + (-8.36e7 - 3.04e7i)T + (2.68e17 + 2.25e17i)T^{2} \) |
| 97 | \( 1 + (1.71e8 + 1.43e8i)T + (1.32e17 + 7.48e17i)T^{2} \) |
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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
−13.70523169313983706575583017251, −13.54992307645523299783106104551, −12.41215724172721404946594030792, −10.05974517129254226605728621411, −8.772142743646762627691701086280, −7.62076310993105864918634277041, −6.63421824709330284656123271968, −4.13279685082254714145159454926, −3.09976653213793398015116404876, −1.16020439340116357731917490928,
2.44184710826711399159807222619, 2.89058567423592656799714720732, 4.50606792488143912426984962420, 6.54210102847552117146754017704, 8.563613686524129048711165099092, 9.406043289509872215157160526015, 10.51910082838648794573807861471, 12.38478247327023634135599803456, 13.36882973612795489278230072620, 14.68773298196038400633244463982
