| L(s) = 1 | + (−8 − 13.8i)2-s + (−122. + 212. i)3-s + (−127. + 221. i)4-s + (1.33e3 + 2.30e3i)5-s + 3.92e3·6-s + 4.09e3·8-s + (−2.02e4 − 3.50e4i)9-s + (2.13e4 − 3.68e4i)10-s + (−3.16e4 + 5.48e4i)11-s + (−3.13e4 − 5.43e4i)12-s + 1.47e3·13-s − 6.52e5·15-s + (−3.27e4 − 5.67e4i)16-s + (−1.00e5 + 1.73e5i)17-s + (−3.23e5 + 5.60e5i)18-s + (−2.30e5 − 3.98e5i)19-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.873 + 1.51i)3-s + (−0.249 + 0.433i)4-s + (0.952 + 1.64i)5-s + 1.23·6-s + 0.353·8-s + (−1.02 − 1.77i)9-s + (0.673 − 1.16i)10-s + (−0.651 + 1.12i)11-s + (−0.436 − 0.756i)12-s + 0.0143·13-s − 3.32·15-s + (−0.125 − 0.216i)16-s + (−0.290 + 0.503i)17-s + (−0.726 + 1.25i)18-s + (−0.405 − 0.701i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.324022 - 0.396043i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.324022 - 0.396043i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (8 + 13.8i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (122. - 212. i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-1.33e3 - 2.30e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (3.16e4 - 5.48e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 - 1.47e3T + 1.06e10T^{2} \) |
| 17 | \( 1 + (1.00e5 - 1.73e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (2.30e5 + 3.98e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-1.63e5 - 2.82e5i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 - 1.62e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (1.47e6 - 2.56e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (9.66e6 + 1.67e7i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 - 4.07e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.34e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-3.01e7 - 5.22e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (2.35e7 - 4.08e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (1.44e7 - 2.50e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (1.56e7 + 2.70e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-2.38e7 + 4.12e7i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 - 2.00e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (-1.43e8 + 2.48e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-3.26e7 - 5.65e7i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + 5.45e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (2.48e7 + 4.30e7i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 - 7.59e8T + 7.60e17T^{2} \) |
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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
−12.57608286174442410549577977533, −11.14223490819449943732212570791, −10.66717785162747863853436885434, −10.02419024830683451143719099120, −9.214146558526359968445495975774, −7.13345345721951906623493095574, −5.93586668728533196561182890798, −4.68658845573259011560261615770, −3.34406583067488182284647165790, −2.16541105714397656309150676485,
0.20172439738112090075247442256, 0.963207107920922601663853936316, 1.99987136520512061517751977904, 5.04459946680302934475528418766, 5.71350785558534139678054067520, 6.61080043303553849022374896912, 8.081450181572159505644624489847, 8.687983494663329953740820498272, 10.17469421287946086832682088363, 11.60206177484698130070551627918
