| L(s) = 1 | + (−0.212 + 0.977i)2-s + (0.800 − 0.599i)3-s + (−0.909 − 0.415i)4-s + (−1.64 − 1.51i)5-s + (0.415 + 0.909i)6-s + (−0.129 + 1.80i)7-s + (0.599 − 0.800i)8-s + (0.281 − 0.959i)9-s + (1.83 − 1.28i)10-s + (−2.56 + 3.98i)11-s + (−0.977 + 0.212i)12-s + (−4.05 + 0.290i)13-s + (−1.73 − 0.510i)14-s + (−2.22 − 0.228i)15-s + (0.654 + 0.755i)16-s + (−0.977 + 2.62i)17-s + ⋯ |
| L(s) = 1 | + (−0.150 + 0.690i)2-s + (0.462 − 0.345i)3-s + (−0.454 − 0.207i)4-s + (−0.735 − 0.678i)5-s + (0.169 + 0.371i)6-s + (−0.0488 + 0.683i)7-s + (0.211 − 0.283i)8-s + (0.0939 − 0.319i)9-s + (0.578 − 0.405i)10-s + (−0.772 + 1.20i)11-s + (−0.282 + 0.0613i)12-s + (−1.12 + 0.0804i)13-s + (−0.464 − 0.136i)14-s + (−0.574 − 0.0590i)15-s + (0.163 + 0.188i)16-s + (−0.237 + 0.635i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.378i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.106195 + 0.539666i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.106195 + 0.539666i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.212 - 0.977i)T \) |
| 3 | \( 1 + (-0.800 + 0.599i)T \) |
| 5 | \( 1 + (1.64 + 1.51i)T \) |
| 23 | \( 1 + (-1.29 - 4.61i)T \) |
| good | 7 | \( 1 + (0.129 - 1.80i)T + (-6.92 - 0.996i)T^{2} \) |
| 11 | \( 1 + (2.56 - 3.98i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (4.05 - 0.290i)T + (12.8 - 1.85i)T^{2} \) |
| 17 | \( 1 + (0.977 - 2.62i)T + (-12.8 - 11.1i)T^{2} \) |
| 19 | \( 1 + (-1.91 + 4.20i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (4.86 - 2.21i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-1.46 - 10.2i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (3.08 + 5.65i)T + (-20.0 + 31.1i)T^{2} \) |
| 41 | \( 1 + (3.86 - 1.13i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (1.11 + 1.48i)T + (-12.1 + 41.2i)T^{2} \) |
| 47 | \( 1 + (8.07 - 8.07i)T - 47iT^{2} \) |
| 53 | \( 1 + (6.89 + 0.492i)T + (52.4 + 7.54i)T^{2} \) |
| 59 | \( 1 + (11.4 + 9.94i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (0.941 - 0.135i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-13.5 - 2.94i)T + (60.9 + 27.8i)T^{2} \) |
| 71 | \( 1 + (-7.51 + 4.83i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (0.171 - 0.0639i)T + (55.1 - 47.8i)T^{2} \) |
| 79 | \( 1 + (-10.3 + 11.9i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-3.56 + 1.94i)T + (44.8 - 69.8i)T^{2} \) |
| 89 | \( 1 + (-1.36 + 9.50i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (1.81 + 0.990i)T + (52.4 + 81.6i)T^{2} \) |
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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
−10.75860663112539366453977048935, −9.499565541882570305343810743389, −9.125860885160911948732996536158, −8.066966898592608949324678133711, −7.48211006692577394454535159686, −6.74725613511893269672428551092, −5.19104634080463167720860313273, −4.81330484510434417821061274378, −3.31063696632036758466330934817, −1.85154518596557161515687228110,
0.27581412455224976000401314514, 2.47527173212689687422881660741, 3.31096007646026799584196301129, 4.18525165484435765940361947403, 5.29330319473245697070274376752, 6.75679483495034397548242262237, 7.84837474393113544783726249807, 8.168281514036420970214243236505, 9.510340865750861938545768689333, 10.16587829269874831181841642402
