L(s) = 1 | + (−0.474 − 0.127i)2-s + (−1.40 − 1.01i)3-s + (−1.52 − 0.879i)4-s + (2.23 + 0.0580i)5-s + (0.536 + 0.659i)6-s + (1.30 + 1.30i)8-s + (0.942 + 2.84i)9-s + (−1.05 − 0.311i)10-s + (−2.31 − 1.33i)11-s + (1.24 + 2.78i)12-s + (−2.14 + 2.14i)13-s + (−3.07 − 2.34i)15-s + (1.30 + 2.26i)16-s + (1.19 + 4.46i)17-s + (−0.0850 − 1.46i)18-s + (−4.54 + 2.62i)19-s + ⋯ |
L(s) = 1 | + (−0.335 − 0.0898i)2-s + (−0.810 − 0.585i)3-s + (−0.761 − 0.439i)4-s + (0.999 + 0.0259i)5-s + (0.219 + 0.269i)6-s + (0.461 + 0.461i)8-s + (0.314 + 0.949i)9-s + (−0.332 − 0.0984i)10-s + (−0.697 − 0.402i)11-s + (0.359 + 0.802i)12-s + (−0.596 + 0.596i)13-s + (−0.795 − 0.606i)15-s + (0.326 + 0.565i)16-s + (0.290 + 1.08i)17-s + (−0.0200 − 0.346i)18-s + (−1.04 + 0.601i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 - 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.767 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.632862 + 0.229473i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.632862 + 0.229473i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.40 + 1.01i)T \) |
| 5 | \( 1 + (-2.23 - 0.0580i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.474 + 0.127i)T + (1.73 + i)T^{2} \) |
| 11 | \( 1 + (2.31 + 1.33i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.14 - 2.14i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.19 - 4.46i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (4.54 - 2.62i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.932 + 3.48i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 2.86T + 29T^{2} \) |
| 31 | \( 1 + (2.64 - 4.57i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.784 - 2.92i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 11.5iT - 41T^{2} \) |
| 43 | \( 1 + (-0.759 + 0.759i)T - 43iT^{2} \) |
| 47 | \( 1 + (-10.4 - 2.80i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-6.05 + 1.62i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (0.0797 - 0.138i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.36 - 4.09i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.39 + 1.98i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 13.5iT - 71T^{2} \) |
| 73 | \( 1 + (1.52 + 5.68i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-3.37 + 1.94i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.03 - 4.03i)T + 83iT^{2} \) |
| 89 | \( 1 + (1.97 + 3.42i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.86 - 1.86i)T + 97iT^{2} \) |
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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
−10.44541424230319166305894404254, −9.870391191780233014780498197648, −8.723563917196302009212712929042, −8.074161882496459261870955728837, −6.76817216248348668410628363369, −5.98892776840875801719217283017, −5.26640516156699466955119151996, −4.36280125942381222801201253328, −2.34775849665219221888057991569, −1.25028356242382184721289549713,
0.48741496409871491527976639960, 2.55387972350686914970291895819, 3.99229614888568851069839927328, 5.10027381185073964658744336376, 5.46956585899154530816822285194, 6.83573044754860290421093273206, 7.63382155738293831175521103627, 8.955339573740337012445624793553, 9.435466438414381942752983795241, 10.24288013606504393030326011392