| L(s) = 1 | + (−0.218 − 0.00879i)2-s + (−0.278 + 0.960i)3-s + (−1.94 − 0.157i)4-s + (−3.83 − 1.45i)5-s + (0.0691 − 0.207i)6-s + (1.70 + 4.00i)7-s + (0.857 + 0.104i)8-s + (−0.845 − 0.534i)9-s + (0.824 + 0.351i)10-s + (−0.471 − 0.745i)11-s + (0.692 − 1.82i)12-s + (0.0746 − 3.60i)13-s + (−0.337 − 0.890i)14-s + (2.46 − 3.27i)15-s + (3.66 + 0.596i)16-s + (−2.47 + 1.05i)17-s + ⋯ |
| L(s) = 1 | + (−0.154 − 0.00622i)2-s + (−0.160 + 0.554i)3-s + (−0.972 − 0.0785i)4-s + (−1.71 − 0.650i)5-s + (0.0282 − 0.0846i)6-s + (0.645 + 1.51i)7-s + (0.303 + 0.0368i)8-s + (−0.281 − 0.178i)9-s + (0.260 + 0.111i)10-s + (−0.142 − 0.224i)11-s + (0.199 − 0.526i)12-s + (0.0206 − 0.999i)13-s + (−0.0902 − 0.237i)14-s + (0.635 − 0.846i)15-s + (0.916 + 0.149i)16-s + (−0.601 + 0.256i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.703 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.532585 - 0.222405i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.532585 - 0.222405i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.278 - 0.960i)T \) |
| 13 | \( 1 + (-0.0746 + 3.60i)T \) |
| good | 2 | \( 1 + (0.218 + 0.00879i)T + (1.99 + 0.160i)T^{2} \) |
| 5 | \( 1 + (3.83 + 1.45i)T + (3.74 + 3.31i)T^{2} \) |
| 7 | \( 1 + (-1.70 - 4.00i)T + (-4.84 + 5.04i)T^{2} \) |
| 11 | \( 1 + (0.471 + 0.745i)T + (-4.71 + 9.93i)T^{2} \) |
| 17 | \( 1 + (2.47 - 1.05i)T + (11.7 - 12.2i)T^{2} \) |
| 19 | \( 1 + (-5.33 + 3.08i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.74 + 4.75i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.160 + 3.97i)T + (-28.9 - 2.33i)T^{2} \) |
| 31 | \( 1 + (5.18 + 5.84i)T + (-3.73 + 30.7i)T^{2} \) |
| 37 | \( 1 + (-1.36 + 0.277i)T + (34.0 - 14.5i)T^{2} \) |
| 41 | \( 1 + (-8.14 - 2.35i)T + (34.6 + 21.9i)T^{2} \) |
| 43 | \( 1 + (0.847 - 4.15i)T + (-39.5 - 16.8i)T^{2} \) |
| 47 | \( 1 + (1.15 - 0.795i)T + (16.6 - 43.9i)T^{2} \) |
| 53 | \( 1 + (0.436 - 3.59i)T + (-51.4 - 12.6i)T^{2} \) |
| 59 | \( 1 + (1.39 + 8.59i)T + (-55.9 + 18.6i)T^{2} \) |
| 61 | \( 1 + (-9.91 + 7.44i)T + (16.9 - 58.5i)T^{2} \) |
| 67 | \( 1 + (-0.132 - 1.64i)T + (-66.1 + 10.7i)T^{2} \) |
| 71 | \( 1 + (-2.58 + 2.48i)T + (2.85 - 70.9i)T^{2} \) |
| 73 | \( 1 + (3.89 - 7.41i)T + (-41.4 - 60.0i)T^{2} \) |
| 79 | \( 1 + (1.09 + 1.58i)T + (-28.0 + 73.8i)T^{2} \) |
| 83 | \( 1 + (-1.12 + 4.57i)T + (-73.4 - 38.5i)T^{2} \) |
| 89 | \( 1 + (12.4 + 7.17i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (13.8 + 11.2i)T + (19.4 + 95.0i)T^{2} \) |
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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
−11.15569084718601796642844839228, −9.651129218331567209987736016831, −8.857074532649146671358639990238, −8.318345763817748682252970558509, −7.65420522331906345786312940057, −5.72055537592881344187564072122, −4.95672379294748821721054009569, −4.27149956778340065581649520410, −3.02730416327316151518749423218, −0.49762895392004615036615818624,
1.08719015087849619984228899215, 3.50167610109147189345934546409, 4.14549514070096291246840223830, 5.10282067939758399006454500849, 7.13552087113819421922945970119, 7.26327055217090304960442367221, 8.083832679185544764905827671675, 9.081067635001118940263447865953, 10.34948432591799877253493319350, 11.11711593484659555031063044988
