| L(s) = 1 | + (−1.22 + 0.0642i)2-s + (0.158 − 0.195i)3-s + (−0.490 + 0.0515i)4-s + (−2.14 − 0.624i)5-s + (−0.181 + 0.249i)6-s + (2.13 + 1.56i)7-s + (3.02 − 0.478i)8-s + (0.610 + 2.87i)9-s + (2.67 + 0.627i)10-s + (6.21 + 1.32i)11-s + (−0.0675 + 0.104i)12-s + (−3.08 + 1.57i)13-s + (−2.71 − 1.77i)14-s + (−0.462 + 0.321i)15-s + (−2.71 + 0.576i)16-s + (1.38 − 3.61i)17-s + ⋯ |
| L(s) = 1 | + (−0.866 + 0.0454i)2-s + (0.0914 − 0.112i)3-s + (−0.245 + 0.0257i)4-s + (−0.960 − 0.279i)5-s + (−0.0741 + 0.102i)6-s + (0.806 + 0.590i)7-s + (1.06 − 0.169i)8-s + (0.203 + 0.957i)9-s + (0.845 + 0.198i)10-s + (1.87 + 0.398i)11-s + (−0.0195 + 0.0300i)12-s + (−0.856 + 0.436i)13-s + (−0.726 − 0.475i)14-s + (−0.119 + 0.0828i)15-s + (−0.677 + 0.144i)16-s + (0.336 − 0.875i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.623413 + 0.253186i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.623413 + 0.253186i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.14 + 0.624i)T \) |
| 7 | \( 1 + (-2.13 - 1.56i)T \) |
| good | 2 | \( 1 + (1.22 - 0.0642i)T + (1.98 - 0.209i)T^{2} \) |
| 3 | \( 1 + (-0.158 + 0.195i)T + (-0.623 - 2.93i)T^{2} \) |
| 11 | \( 1 + (-6.21 - 1.32i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (3.08 - 1.57i)T + (7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-1.38 + 3.61i)T + (-12.6 - 11.3i)T^{2} \) |
| 19 | \( 1 + (0.630 - 6.00i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (0.0931 + 1.77i)T + (-22.8 + 2.40i)T^{2} \) |
| 29 | \( 1 + (-1.19 - 1.63i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.613 - 1.37i)T + (-20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (6.46 + 4.19i)T + (15.0 + 33.8i)T^{2} \) |
| 41 | \( 1 + (-2.44 - 0.795i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-4.23 + 4.23i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.45 - 0.941i)T + (34.9 - 31.4i)T^{2} \) |
| 53 | \( 1 + (-2.37 - 1.91i)T + (11.0 + 51.8i)T^{2} \) |
| 59 | \( 1 + (8.64 + 9.59i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (0.0639 + 0.0575i)T + (6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (1.23 + 0.475i)T + (49.7 + 44.8i)T^{2} \) |
| 71 | \( 1 + (-1.14 + 0.833i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-7.23 - 11.1i)T + (-29.6 + 66.6i)T^{2} \) |
| 79 | \( 1 + (-1.79 + 4.03i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-0.0358 - 0.226i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-8.63 + 9.59i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (-0.0559 + 0.353i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−12.44475787895120183445323575712, −11.90945042496636084268823857501, −10.80305985826987898480929519419, −9.553148650646994500044868094296, −8.746331846680598457467746178273, −7.87485430651206341777300810742, −7.07924097901248467716583487478, −5.00874953678403521393653338676, −4.11782992855211580102522020655, −1.63845506802907923426566309168,
0.988986092425500048953726510190, 3.72126772053863369333731827380, 4.59604848344746306246005428209, 6.66729095789841649998106046342, 7.62446494634888174701025908356, 8.623531008189793896455749607390, 9.421270731277893155482004210324, 10.54775267805287994852209335954, 11.46229545911240961169533417458, 12.28018625301323317406708461383
