| L(s) = 1 | + (1.73 − 5.34i)2-s + (−11.2 + 3.64i)3-s + (−12.5 − 9.14i)4-s − 23.2·5-s + 66.3i·6-s + (−34.4 − 25.0i)7-s + (2.03 − 1.47i)8-s + (47.1 − 34.2i)9-s + (−40.3 + 124. i)10-s + (35.6 − 49.1i)11-s + (174. + 56.7i)12-s + (−21.3 + 6.95i)13-s + (−193. + 140. i)14-s + (260. − 84.6i)15-s + (−81.2 − 250. i)16-s + (−302. − 416. i)17-s + ⋯ |
| L(s) = 1 | + (0.433 − 1.33i)2-s + (−1.24 + 0.405i)3-s + (−0.786 − 0.571i)4-s − 0.928·5-s + 1.84i·6-s + (−0.703 − 0.511i)7-s + (0.0318 − 0.0231i)8-s + (0.582 − 0.423i)9-s + (−0.403 + 1.24i)10-s + (0.295 − 0.406i)11-s + (1.21 + 0.393i)12-s + (−0.126 + 0.0411i)13-s + (−0.987 + 0.717i)14-s + (1.15 − 0.376i)15-s + (−0.317 − 0.976i)16-s + (−1.04 − 1.44i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 - 0.429i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.903 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.107543 + 0.476382i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.107543 + 0.476382i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (818. - 503. i)T \) |
| good | 2 | \( 1 + (-1.73 + 5.34i)T + (-12.9 - 9.40i)T^{2} \) |
| 3 | \( 1 + (11.2 - 3.64i)T + (65.5 - 47.6i)T^{2} \) |
| 5 | \( 1 + 23.2T + 625T^{2} \) |
| 7 | \( 1 + (34.4 + 25.0i)T + (741. + 2.28e3i)T^{2} \) |
| 11 | \( 1 + (-35.6 + 49.1i)T + (-4.52e3 - 1.39e4i)T^{2} \) |
| 13 | \( 1 + (21.3 - 6.95i)T + (2.31e4 - 1.67e4i)T^{2} \) |
| 17 | \( 1 + (302. + 416. i)T + (-2.58e4 + 7.94e4i)T^{2} \) |
| 19 | \( 1 + (23.4 - 72.0i)T + (-1.05e5 - 7.66e4i)T^{2} \) |
| 23 | \( 1 + (-547. - 753. i)T + (-8.64e4 + 2.66e5i)T^{2} \) |
| 29 | \( 1 + (-903. - 293. i)T + (5.72e5 + 4.15e5i)T^{2} \) |
| 37 | \( 1 + 2.66e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (442. - 1.36e3i)T + (-2.28e6 - 1.66e6i)T^{2} \) |
| 43 | \( 1 + (875. + 284. i)T + (2.76e6 + 2.00e6i)T^{2} \) |
| 47 | \( 1 + (-191. - 589. i)T + (-3.94e6 + 2.86e6i)T^{2} \) |
| 53 | \( 1 + (2.16e3 + 2.98e3i)T + (-2.43e6 + 7.50e6i)T^{2} \) |
| 59 | \( 1 + (-132. - 406. i)T + (-9.80e6 + 7.12e6i)T^{2} \) |
| 61 | \( 1 + 4.50e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 2.03e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (-5.47e3 + 3.97e3i)T + (7.85e6 - 2.41e7i)T^{2} \) |
| 73 | \( 1 + (-734. + 1.01e3i)T + (-8.77e6 - 2.70e7i)T^{2} \) |
| 79 | \( 1 + (-3.52e3 - 4.85e3i)T + (-1.20e7 + 3.70e7i)T^{2} \) |
| 83 | \( 1 + (729. + 237. i)T + (3.83e7 + 2.78e7i)T^{2} \) |
| 89 | \( 1 + (1.88e3 - 2.60e3i)T + (-1.93e7 - 5.96e7i)T^{2} \) |
| 97 | \( 1 + (2.61e3 + 1.90e3i)T + (2.73e7 + 8.41e7i)T^{2} \) |
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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
−15.81845951800543249703850514023, −13.76824878025344769218754603369, −12.50995404697132022255932754144, −11.43393881347878796890471038107, −10.98603095234894365366343644625, −9.565952872868310846417873387398, −7.00459246077164842899495870968, −4.91352830292368182493688173925, −3.51692469527941964689960986452, −0.36107024900851938620737703758,
4.56315602828913253554488577840, 6.14841520904974581357203708703, 6.89017057016222954846986754473, 8.490403885203869100234304303469, 10.82553779394351499849796934670, 12.12821492356465267643602299109, 13.07963459611692904208958314947, 14.93707980759653871625191075942, 15.63002400084643028553780547434, 16.75143522410455494574312388826
