| L(s) = 1 | + (−1.88 + 1.16i)2-s + (0.0289 + 0.101i)3-s + (1.29 − 2.61i)4-s + (2.77 − 1.07i)5-s + (−0.173 − 0.157i)6-s + (−0.356 + 3.84i)7-s + (0.187 + 2.02i)8-s + (2.54 − 1.57i)9-s + (−3.98 + 5.27i)10-s + (−2.62 + 1.62i)11-s + (0.302 + 0.0566i)12-s + (−0.512 + 5.53i)13-s + (−3.81 − 7.65i)14-s + (0.189 + 0.251i)15-s + (0.800 + 1.06i)16-s + (4.16 − 3.79i)17-s + ⋯ |
| L(s) = 1 | + (−1.33 + 0.825i)2-s + (0.0166 + 0.0586i)3-s + (0.649 − 1.30i)4-s + (1.24 − 0.481i)5-s + (−0.0706 − 0.0644i)6-s + (−0.134 + 1.45i)7-s + (0.0662 + 0.714i)8-s + (0.847 − 0.524i)9-s + (−1.25 + 1.66i)10-s + (−0.792 + 0.490i)11-s + (0.0874 + 0.0163i)12-s + (−0.142 + 1.53i)13-s + (−1.01 − 2.04i)14-s + (0.0489 + 0.0648i)15-s + (0.200 + 0.265i)16-s + (1.01 − 0.921i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.540496 + 0.375059i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.540496 + 0.375059i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 + (-9.10 + 4.48i)T \) |
| good | 2 | \( 1 + (1.88 - 1.16i)T + (0.891 - 1.79i)T^{2} \) |
| 3 | \( 1 + (-0.0289 - 0.101i)T + (-2.55 + 1.57i)T^{2} \) |
| 5 | \( 1 + (-2.77 + 1.07i)T + (3.69 - 3.36i)T^{2} \) |
| 7 | \( 1 + (0.356 - 3.84i)T + (-6.88 - 1.28i)T^{2} \) |
| 11 | \( 1 + (2.62 - 1.62i)T + (4.90 - 9.84i)T^{2} \) |
| 13 | \( 1 + (0.512 - 5.53i)T + (-12.7 - 2.38i)T^{2} \) |
| 17 | \( 1 + (-4.16 + 3.79i)T + (1.56 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.02 + 3.59i)T + (-16.1 - 10.0i)T^{2} \) |
| 23 | \( 1 + (2.01 + 1.25i)T + (10.2 + 20.5i)T^{2} \) |
| 29 | \( 1 + (-0.818 + 0.316i)T + (21.4 - 19.5i)T^{2} \) |
| 31 | \( 1 + (3.23 + 4.27i)T + (-8.48 + 29.8i)T^{2} \) |
| 37 | \( 1 + (5.95 + 1.11i)T + (34.5 + 13.3i)T^{2} \) |
| 41 | \( 1 + (5.40 + 2.09i)T + (30.2 + 27.6i)T^{2} \) |
| 43 | \( 1 + (-10.1 + 1.88i)T + (40.0 - 15.5i)T^{2} \) |
| 47 | \( 1 + 5.92T + 47T^{2} \) |
| 53 | \( 1 + (-0.351 + 1.23i)T + (-45.0 - 27.9i)T^{2} \) |
| 59 | \( 1 + (0.663 + 7.16i)T + (-57.9 + 10.8i)T^{2} \) |
| 61 | \( 1 + (4.64 - 4.23i)T + (5.62 - 60.7i)T^{2} \) |
| 67 | \( 1 + (0.432 + 4.66i)T + (-65.8 + 12.3i)T^{2} \) |
| 71 | \( 1 + (-1.70 - 0.661i)T + (52.4 + 47.8i)T^{2} \) |
| 73 | \( 1 + (10.6 + 4.11i)T + (53.9 + 49.1i)T^{2} \) |
| 79 | \( 1 + (0.587 - 0.227i)T + (58.3 - 53.2i)T^{2} \) |
| 83 | \( 1 + (0.388 - 4.19i)T + (-81.5 - 15.2i)T^{2} \) |
| 89 | \( 1 + (0.831 + 1.66i)T + (-53.6 + 71.0i)T^{2} \) |
| 97 | \( 1 + (-6.02 - 5.48i)T + (8.95 + 96.5i)T^{2} \) |
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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
−14.28202014152157492060401240756, −12.94479499915833196596844876131, −11.93782005059359060708994861628, −10.11083493388715638315889817574, −9.358548429971692623636296790678, −8.991789468710150606016711787558, −7.39383651960150843972042833824, −6.28748959981572515306437361958, −5.17023219957387587586437855897, −1.98799580916901001405801701294,
1.46457362703087941672164980556, 3.22515339420498544387835011683, 5.62749320649766841296260291361, 7.38812668774910325445526562007, 8.127441558284066926322528088317, 9.919448652760569678771890591729, 10.35910950073217693029074249441, 10.65579401486292035537933282268, 12.55929848531840755071532497189, 13.42736957489844708027350053844
