| L(s) = 1 | + (12.6 − 21.9i)3-s + (−1.44e3 + 837. i)5-s + (5.64e3 + 2.91e3i)7-s + (9.51e3 + 1.64e4i)9-s + (−3.27e4 − 1.88e4i)11-s + 6.72e4i·13-s + 4.24e4i·15-s + (4.08e5 + 2.35e5i)17-s + (−2.07e5 − 3.60e5i)19-s + (1.35e5 − 8.70e4i)21-s + (1.49e6 − 8.61e5i)23-s + (4.24e5 − 7.35e5i)25-s + 9.82e5·27-s − 5.61e6·29-s + (−2.31e6 + 4.00e6i)31-s + ⋯ |
| L(s) = 1 | + (0.0904 − 0.156i)3-s + (−1.03 + 0.598i)5-s + (0.888 + 0.458i)7-s + (0.483 + 0.837i)9-s + (−0.673 − 0.389i)11-s + 0.652i·13-s + 0.216i·15-s + (1.18 + 0.685i)17-s + (−0.365 − 0.633i)19-s + (0.152 − 0.0976i)21-s + (1.11 − 0.641i)23-s + (0.217 − 0.376i)25-s + 0.355·27-s − 1.47·29-s + (−0.450 + 0.779i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.9889708207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9889708207\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-5.64e3 - 2.91e3i)T \) |
| good | 3 | \( 1 + (-12.6 + 21.9i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (1.44e3 - 837. i)T + (9.76e5 - 1.69e6i)T^{2} \) |
| 11 | \( 1 + (3.27e4 + 1.88e4i)T + (1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 - 6.72e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 + (-4.08e5 - 2.35e5i)T + (5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (2.07e5 + 3.60e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-1.49e6 + 8.61e5i)T + (9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + 5.61e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (2.31e6 - 4.00e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (4.77e5 + 8.26e5i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 + 5.75e6iT - 3.27e14T^{2} \) |
| 43 | \( 1 + 5.01e5iT - 5.02e14T^{2} \) |
| 47 | \( 1 + (-6.09e6 - 1.05e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-2.17e6 + 3.77e6i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (2.04e7 - 3.53e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (1.27e8 - 7.33e7i)T + (5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (1.91e8 + 1.10e8i)T + (1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 - 3.38e8iT - 4.58e16T^{2} \) |
| 73 | \( 1 + (3.86e8 + 2.23e8i)T + (2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (4.21e8 - 2.43e8i)T + (5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 - 6.31e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (5.18e8 - 2.99e8i)T + (1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + 1.18e9iT - 7.60e17T^{2} \) |
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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
−12.21041629789475055881067422071, −11.12198445524499367034273620419, −10.60811301158545842050857937378, −8.856715338826409889981727050389, −7.84701666246492439281586720325, −7.17431073187662511390734592924, −5.46939352776767624312234334259, −4.30164153843872407937562185193, −2.89746444391669280386929117644, −1.54494650733042897238668728672,
0.26201692118895746161658719936, 1.36567159299456073007334917062, 3.36060761587934145342273144399, 4.39040093690299011957836283977, 5.46208964560840719638314937014, 7.41793635103239162719479138102, 7.87440245571548253483397583640, 9.199153053286977227099913549227, 10.35454379468773541715497586008, 11.48984544653732690568035855754
