| L(s) = 1 | + (−0.766 + 0.642i)2-s + (0.939 + 0.342i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (−0.939 + 0.342i)6-s + (0.266 − 0.460i)7-s + (0.500 + 0.866i)8-s + (0.766 + 0.642i)9-s + (0.766 + 0.642i)10-s + (−1.85 − 3.21i)11-s + (0.499 − 0.866i)12-s + (2.09 − 0.761i)13-s + (0.0923 + 0.524i)14-s + (0.173 − 0.984i)15-s + (−0.939 − 0.342i)16-s + (2.43 − 2.04i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (0.542 + 0.197i)3-s + (0.0868 − 0.492i)4-s + (−0.0776 − 0.440i)5-s + (−0.383 + 0.139i)6-s + (0.100 − 0.174i)7-s + (0.176 + 0.306i)8-s + (0.255 + 0.214i)9-s + (0.242 + 0.203i)10-s + (−0.560 − 0.970i)11-s + (0.144 − 0.250i)12-s + (0.580 − 0.211i)13-s + (0.0246 + 0.140i)14-s + (0.0448 − 0.254i)15-s + (−0.234 − 0.0855i)16-s + (0.591 − 0.496i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 + 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30683 - 0.200615i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30683 - 0.200615i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 3 | \( 1 + (-0.939 - 0.342i)T \) |
| 5 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 + (-3.5 - 2.59i)T \) |
| good | 7 | \( 1 + (-0.266 + 0.460i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.85 + 3.21i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.09 + 0.761i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.43 + 2.04i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.865 + 4.90i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (2.67 + 2.24i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.52 + 2.63i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 5.71T + 37T^{2} \) |
| 41 | \( 1 + (-3.70 - 1.34i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.44 + 8.18i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (0.754 + 0.633i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (0.144 - 0.817i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (7.72 - 6.48i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (0.774 - 4.39i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-8.41 - 7.06i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.127 - 0.725i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (5.16 + 1.87i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-13.1 - 4.80i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-2.31 + 4.01i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.80 - 1.02i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (3.01 - 2.52i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.51909213245280968498382456260, −9.667400040810178025248233111273, −8.824062152565491036244756500387, −8.078721009570691406831256736321, −7.47499618005500938178309970623, −6.09993989294394887450847555026, −5.29228264960409890063957582794, −4.01178984363588441796927598759, −2.74381254230467785679306032049, −0.951181918214195893670420141669,
1.53963783770125931891956233506, 2.76565060929595089852085674446, 3.75083995730667298725444205468, 5.12095929337421819705732010965, 6.51450845322825889046641689489, 7.51874769504818010493267595815, 8.037178732644437155614353072905, 9.228922294198463132910411129212, 9.758604071900864043271289411225, 10.75175819270950001209557381396
