| L(s) = 1 | + (0.883 + 0.910i)2-s + (−1.90 − 2.52i)3-s + (0.0120 − 0.389i)4-s + (−0.470 − 1.33i)5-s + (0.616 − 3.96i)6-s + (−0.262 + 1.19i)7-s + (2.24 − 2.04i)8-s + (−1.92 + 6.75i)9-s + (0.801 − 1.60i)10-s + (2.89 − 0.726i)11-s + (−1.00 + 0.712i)12-s + (1.81 + 1.65i)13-s + (−1.32 + 0.818i)14-s + (−2.47 + 3.73i)15-s + (3.06 + 0.188i)16-s + (0.554 + 3.57i)17-s + ⋯ |
| L(s) = 1 | + (0.624 + 0.644i)2-s + (−1.10 − 1.45i)3-s + (0.00600 − 0.194i)4-s + (−0.210 − 0.597i)5-s + (0.251 − 1.62i)6-s + (−0.0990 + 0.452i)7-s + (0.792 − 0.722i)8-s + (−0.640 + 2.25i)9-s + (0.253 − 0.508i)10-s + (0.871 − 0.219i)11-s + (−0.290 + 0.205i)12-s + (0.503 + 0.459i)13-s + (−0.353 + 0.218i)14-s + (−0.639 + 0.965i)15-s + (0.765 + 0.0472i)16-s + (0.134 + 0.865i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.869432 - 0.513155i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.869432 - 0.513155i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 + (5.21 - 8.70i)T \) |
| good | 2 | \( 1 + (-0.883 - 0.910i)T + (-0.0615 + 1.99i)T^{2} \) |
| 3 | \( 1 + (1.90 + 2.52i)T + (-0.820 + 2.88i)T^{2} \) |
| 5 | \( 1 + (0.470 + 1.33i)T + (-3.89 + 3.13i)T^{2} \) |
| 7 | \( 1 + (0.262 - 1.19i)T + (-6.35 - 2.92i)T^{2} \) |
| 11 | \( 1 + (-2.89 + 0.726i)T + (9.69 - 5.20i)T^{2} \) |
| 13 | \( 1 + (-1.81 - 1.65i)T + (1.19 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.554 - 3.57i)T + (-16.2 + 5.15i)T^{2} \) |
| 19 | \( 1 + (2.69 + 6.37i)T + (-13.2 + 13.6i)T^{2} \) |
| 23 | \( 1 + (-1.69 - 5.94i)T + (-19.5 + 12.1i)T^{2} \) |
| 29 | \( 1 + (1.96 - 2.29i)T + (-4.44 - 28.6i)T^{2} \) |
| 31 | \( 1 + (3.90 + 7.83i)T + (-18.6 + 24.7i)T^{2} \) |
| 37 | \( 1 + (-0.289 - 3.12i)T + (-36.3 + 6.79i)T^{2} \) |
| 41 | \( 1 + (3.68 - 10.4i)T + (-31.9 - 25.7i)T^{2} \) |
| 43 | \( 1 + (-4.02 - 2.84i)T + (14.2 + 40.5i)T^{2} \) |
| 47 | \( 1 + (-3.87 - 6.70i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.59 + 0.816i)T + (51.3 + 12.9i)T^{2} \) |
| 59 | \( 1 + (-0.0886 - 0.404i)T + (-53.5 + 24.6i)T^{2} \) |
| 61 | \( 1 + (2.53 + 0.983i)T + (45.0 + 41.0i)T^{2} \) |
| 67 | \( 1 + (1.01 + 0.323i)T + (54.6 + 38.7i)T^{2} \) |
| 71 | \( 1 + (5.30 + 6.18i)T + (-10.8 + 70.1i)T^{2} \) |
| 73 | \( 1 + (7.28 - 1.36i)T + (68.0 - 26.3i)T^{2} \) |
| 79 | \( 1 + (6.55 + 1.22i)T + (73.6 + 28.5i)T^{2} \) |
| 83 | \( 1 + (4.54 - 1.44i)T + (67.7 - 47.9i)T^{2} \) |
| 89 | \( 1 + (-15.7 + 9.76i)T + (39.6 - 79.6i)T^{2} \) |
| 97 | \( 1 + (-2.53 + 16.3i)T + (-92.4 - 29.4i)T^{2} \) |
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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
−13.26765485753894610340754781785, −12.93956778765463460757747743589, −11.72912516428831787518466303163, −10.97172649859489041540253894052, −9.053126198463957703933757071035, −7.59661559404603989478092986101, −6.48254202286309136236483218762, −5.88825087255304587628866189502, −4.61896568318603733659501874194, −1.35989966562290487101896157606,
3.46555055557147840303571155076, 4.21329557375069618502567192612, 5.51502805286337252772634956791, 6.98342024612541130703127684489, 8.882029026682251735883619899030, 10.44184241780038464590600697352, 10.71583020681302025017913667907, 11.80639374997410950230980404792, 12.51089193959315109659082240720, 14.13597575678718194782006692970
