| L(s) = 1 | + (0.173 + 0.984i)2-s + (0.131 + 0.187i)3-s + (−0.939 + 0.342i)4-s + (0.561 − 2.16i)5-s + (−0.162 + 0.162i)6-s + (2.06 − 0.180i)7-s + (−0.5 − 0.866i)8-s + (1.00 − 2.76i)9-s + (2.22 + 0.176i)10-s + (−3.73 + 2.15i)11-s + (−0.187 − 0.131i)12-s + (5.88 − 2.14i)13-s + (0.536 + 2.00i)14-s + (0.480 − 0.179i)15-s + (0.766 − 0.642i)16-s + (0.489 − 1.34i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (0.0759 + 0.108i)3-s + (−0.469 + 0.171i)4-s + (0.251 − 0.967i)5-s + (−0.0662 + 0.0662i)6-s + (0.780 − 0.0682i)7-s + (−0.176 − 0.306i)8-s + (0.336 − 0.923i)9-s + (0.704 + 0.0559i)10-s + (−1.12 + 0.649i)11-s + (−0.0542 − 0.0379i)12-s + (1.63 − 0.594i)13-s + (0.143 + 0.535i)14-s + (0.124 − 0.0463i)15-s + (0.191 − 0.160i)16-s + (0.118 − 0.326i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.57502 + 0.145723i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.57502 + 0.145723i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 5 | \( 1 + (-0.561 + 2.16i)T \) |
| 37 | \( 1 + (5.76 - 1.94i)T \) |
| good | 3 | \( 1 + (-0.131 - 0.187i)T + (-1.02 + 2.81i)T^{2} \) |
| 7 | \( 1 + (-2.06 + 0.180i)T + (6.89 - 1.21i)T^{2} \) |
| 11 | \( 1 + (3.73 - 2.15i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-5.88 + 2.14i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.489 + 1.34i)T + (-13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (-2.98 + 2.08i)T + (6.49 - 17.8i)T^{2} \) |
| 23 | \( 1 + (3.67 - 6.36i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-6.26 - 1.67i)T + (25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (-2.78 - 2.78i)T + 31iT^{2} \) |
| 41 | \( 1 + (-2.57 - 7.07i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + 4.33T + 43T^{2} \) |
| 47 | \( 1 + (2.93 + 10.9i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (4.91 + 0.429i)T + (52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (5.69 + 0.498i)T + (58.1 + 10.2i)T^{2} \) |
| 61 | \( 1 + (-6.74 - 3.14i)T + (39.2 + 46.7i)T^{2} \) |
| 67 | \( 1 + (0.254 + 2.90i)T + (-65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (1.06 - 6.06i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-9.46 + 9.46i)T - 73iT^{2} \) |
| 79 | \( 1 + (0.381 + 4.36i)T + (-77.7 + 13.7i)T^{2} \) |
| 83 | \( 1 + (-4.02 - 8.62i)T + (-53.3 + 63.5i)T^{2} \) |
| 89 | \( 1 + (0.438 - 5.01i)T + (-87.6 - 15.4i)T^{2} \) |
| 97 | \( 1 + (16.2 + 9.38i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.59425328687001441500248504785, −10.28153694517656857056157180548, −9.461653343697758748685350843324, −8.421810787177480957017409337511, −7.891119245380813278561450944195, −6.60126163901690324173620979933, −5.43780096205553809945874377235, −4.76977612662733086987043120231, −3.46476090882600950612453855838, −1.25847388124935346737124186993,
1.74744739262262426614276970078, 2.89533047358028041901993623533, 4.19964567765488842076716814960, 5.46525910299190740233697577607, 6.45953731955880691342617289711, 7.953537769814966656058770882224, 8.384623049883170192929895298059, 9.902967628079341411420372106767, 10.85574652069455669104510006430, 10.93385006376071839314039111900
