| L(s) = 1 | + (−8 − 13.8i)2-s + (39.4 + 68.3i)3-s + (−127. + 221. i)4-s + (−134. − 232. i)5-s + (631. − 1.09e3i)6-s − 5.48e3·7-s + 4.09e3·8-s + (6.72e3 − 1.16e4i)9-s + (−2.14e3 + 3.71e3i)10-s + 1.06e4·11-s − 2.02e4·12-s + (−1.89e4 + 3.28e4i)13-s + (4.38e4 + 7.60e4i)14-s + (1.05e4 − 1.83e4i)15-s + (−3.27e4 − 5.67e4i)16-s + (2.89e5 + 5.00e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.281 + 0.487i)3-s + (−0.249 + 0.433i)4-s + (−0.0959 − 0.166i)5-s + (0.198 − 0.344i)6-s − 0.863·7-s + 0.353·8-s + (0.341 − 0.591i)9-s + (−0.0678 + 0.117i)10-s + 0.220·11-s − 0.281·12-s + (−0.183 + 0.318i)13-s + (0.305 + 0.528i)14-s + (0.0539 − 0.0935i)15-s + (−0.125 − 0.216i)16-s + (0.839 + 1.45i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.57126 - 0.198501i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.57126 - 0.198501i\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + (-5.67e5 - 2.18e4i)T \) |
| good | 3 | \( 1 + (-39.4 - 68.3i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (134. + 232. i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 + 5.48e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 1.06e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (1.89e4 - 3.28e4i)T + (-5.30e9 - 9.18e9i)T^{2} \) |
| 17 | \( 1 + (-2.89e5 - 5.00e5i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 23 | \( 1 + (-2.17e3 + 3.75e3i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (-2.09e6 + 3.63e6i)T + (-7.25e12 - 1.25e13i)T^{2} \) |
| 31 | \( 1 - 8.61e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 3.67e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-1.09e7 - 1.90e7i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 + (-6.66e6 - 1.15e7i)T + (-2.51e14 + 4.35e14i)T^{2} \) |
| 47 | \( 1 + (-5.23e6 + 9.07e6i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (1.47e7 - 2.54e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-3.67e6 - 6.37e6i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-8.61e7 + 1.49e8i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (1.22e8 - 2.12e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (1.00e8 + 1.74e8i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 + (-8.49e7 - 1.47e8i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-2.02e8 - 3.51e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 - 1.99e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-1.82e7 + 3.16e7i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + (6.21e8 + 1.07e9i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
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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
−14.21753974047690729299940413121, −12.77958649729528947416695650078, −11.90171877819904489936029535185, −10.20221807732239643729350422108, −9.552113871197708228584506066355, −8.204894124629751082724049669506, −6.40530432801362804860676265190, −4.24686656316559055257829758793, −3.04431565981528941228599908322, −0.979639981733615889774849774992,
0.913940210135541472552115973922, 2.97144573903812880503632611708, 5.16622174918884311213586068144, 6.84358164451676088106392408857, 7.69029974682206206907151306776, 9.230352089513396872500319299483, 10.34823482634501611953385539976, 12.08871545634523473535614072664, 13.40810209559552589581735147759, 14.26703280692707194190658919799
