| L(s) = 1 | + (1 + 1.73i)2-s + (−1.99 + 3.46i)4-s + (−3 − 5.19i)5-s − 7.99·8-s + (6 − 10.3i)10-s + (15 − 25.9i)11-s − 2·13-s + (−8 − 13.8i)16-s + (−33 + 57.1i)17-s + (−26 − 45.0i)19-s + 24·20-s + 60·22-s + (57 + 98.7i)23-s + (44.5 − 77.0i)25-s + (−2 − 3.46i)26-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.268 − 0.464i)5-s − 0.353·8-s + (0.189 − 0.328i)10-s + (0.411 − 0.712i)11-s − 0.0426·13-s + (−0.125 − 0.216i)16-s + (−0.470 + 0.815i)17-s + (−0.313 − 0.543i)19-s + 0.268·20-s + 0.581·22-s + (0.516 + 0.895i)23-s + (0.355 − 0.616i)25-s + (−0.0150 − 0.0261i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9122373865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9122373865\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 - 1.73i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (3 + 5.19i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-15 + 25.9i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 2T + 2.19e3T^{2} \) |
| 17 | \( 1 + (33 - 57.1i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (26 + 45.0i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-57 - 98.7i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 72T + 2.43e4T^{2} \) |
| 31 | \( 1 + (98 - 169. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-143 - 247. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 378T + 6.89e4T^{2} \) |
| 43 | \( 1 - 164T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-114 - 197. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (174 - 301. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-174 + 301. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (53 + 91.7i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (298 - 516. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 630T + 3.57e5T^{2} \) |
| 73 | \( 1 + (521 - 902. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-44 - 76.2i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.44e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (687 + 1.18e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 34T + 9.12e5T^{2} \) |
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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
−10.06946770712219123274139885853, −8.861101604941372303105554871631, −8.603458514274713797807768888849, −7.52703542281427559816487036403, −6.64804965104437394035672070558, −5.82817501837948010021124025733, −4.85242060712740558978671319614, −4.00390579259311710752307069156, −2.96320253420233940635106291917, −1.30001616275690227137719217520,
0.21270023740601513564615700859, 1.76244568606205722325927727087, 2.79364914463039710885903975291, 3.87562107281846589953842992237, 4.68981066610943421652523364459, 5.75976421998464897104921512349, 6.83186050129593487593997467383, 7.49537530553783123397942145595, 8.755853102314902768029984070298, 9.467618317287535754497551797318
