| L(s) = 1 | + (−4 − 6.92i)2-s + (10.4 − 18.0i)3-s + (−31.9 + 55.4i)4-s + (−108. − 187. i)5-s − 167.·6-s + 511.·8-s + (875. + 1.51e3i)9-s + (−867. + 1.50e3i)10-s + (−1.29e3 + 2.23e3i)11-s + (668. + 1.15e3i)12-s + 8.92e3·13-s − 4.52e3·15-s + (−2.04e3 − 3.54e3i)16-s + (−5.55e3 + 9.62e3i)17-s + (7.00e3 − 1.21e4i)18-s + (−4.32e3 − 7.48e3i)19-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.223 − 0.386i)3-s + (−0.249 + 0.433i)4-s + (−0.387 − 0.671i)5-s − 0.315·6-s + 0.353·8-s + (0.400 + 0.693i)9-s + (−0.274 + 0.474i)10-s + (−0.292 + 0.506i)11-s + (0.111 + 0.193i)12-s + 1.12·13-s − 0.346·15-s + (−0.125 − 0.216i)16-s + (−0.274 + 0.475i)17-s + (0.283 − 0.490i)18-s + (−0.144 − 0.250i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.49839 - 0.742797i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.49839 - 0.742797i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4 + 6.92i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (-10.4 + 18.0i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (108. + 187. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 11 | \( 1 + (1.29e3 - 2.23e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 - 8.92e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + (5.55e3 - 9.62e3i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (4.32e3 + 7.48e3i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-3.30e4 - 5.72e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 - 1.28e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + (1.02e5 - 1.76e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (2.45e5 + 4.25e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 - 6.23e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.22e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (6.02e5 + 1.04e6i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-6.36e5 + 1.10e6i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-8.40e5 + 1.45e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.06e6 - 1.83e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.67e6 + 2.90e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 - 2.49e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (1.63e6 - 2.83e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-2.09e6 - 3.63e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + 3.35e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-1.19e6 - 2.07e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + 4.51e6T + 8.07e13T^{2} \) |
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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
−12.54937427259716296347288385788, −11.23255898023555536583264267689, −10.33992402641659360120307475700, −8.927979960022418593407594917030, −8.166355006709337591074100718995, −7.00309009393652687156485960112, −5.10373977752975428767111457591, −3.81247192311436096413507512212, −2.10962803028631231891697797117, −0.911168422405878722443994046045,
0.839712895403114535226539335440, 3.06311944136038810762847482504, 4.34971298843738378503423515941, 6.05376208239756697813627763977, 7.02111091202473304749395803293, 8.304672243564300997024962273162, 9.253527346906432173796067547198, 10.47247144179597299579000513882, 11.30475441028285365174074463273, 12.81866737486288740412256522852
