| L(s) = 1 | + (0.863 − 2.20i)3-s + (−0.284 − 3.80i)5-s + (−1.30 + 2.26i)7-s + (−1.89 − 1.76i)9-s + (0.456 − 1.99i)11-s + (−1.76 + 1.20i)13-s + (−8.61 − 2.65i)15-s + (0.388 − 5.18i)17-s + (0.603 − 0.559i)19-s + (3.86 + 4.84i)21-s + (2.60 − 0.805i)23-s + (−9.43 + 1.42i)25-s + (0.878 − 0.422i)27-s + (1.43 + 3.65i)29-s + (−7.40 − 1.11i)31-s + ⋯ |
| L(s) = 1 | + (0.498 − 1.27i)3-s + (−0.127 − 1.70i)5-s + (−0.495 + 0.857i)7-s + (−0.632 − 0.586i)9-s + (0.137 − 0.602i)11-s + (−0.490 + 0.334i)13-s + (−2.22 − 0.685i)15-s + (0.0942 − 1.25i)17-s + (0.138 − 0.128i)19-s + (0.842 + 1.05i)21-s + (0.544 − 0.167i)23-s + (−1.88 + 0.284i)25-s + (0.168 − 0.0813i)27-s + (0.266 + 0.679i)29-s + (−1.33 − 0.200i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 + 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.248669 - 1.38773i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.248669 - 1.38773i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + (3.37 - 5.62i)T \) |
| good | 3 | \( 1 + (-0.863 + 2.20i)T + (-2.19 - 2.04i)T^{2} \) |
| 5 | \( 1 + (0.284 + 3.80i)T + (-4.94 + 0.745i)T^{2} \) |
| 7 | \( 1 + (1.30 - 2.26i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.456 + 1.99i)T + (-9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (1.76 - 1.20i)T + (4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-0.388 + 5.18i)T + (-16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (-0.603 + 0.559i)T + (1.41 - 18.9i)T^{2} \) |
| 23 | \( 1 + (-2.60 + 0.805i)T + (19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (-1.43 - 3.65i)T + (-21.2 + 19.7i)T^{2} \) |
| 31 | \( 1 + (7.40 + 1.11i)T + (29.6 + 9.13i)T^{2} \) |
| 37 | \( 1 + (0.431 + 0.746i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.40 - 3.02i)T + (-9.12 - 39.9i)T^{2} \) |
| 47 | \( 1 + (-0.00195 - 0.00857i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (-3.12 - 2.13i)T + (19.3 + 49.3i)T^{2} \) |
| 59 | \( 1 + (-3.54 + 1.70i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (-6.20 + 0.935i)T + (58.2 - 17.9i)T^{2} \) |
| 67 | \( 1 + (-9.35 + 8.67i)T + (5.00 - 66.8i)T^{2} \) |
| 71 | \( 1 + (-0.220 - 0.0680i)T + (58.6 + 39.9i)T^{2} \) |
| 73 | \( 1 + (-9.06 + 6.18i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (-0.133 + 0.231i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.09 + 7.88i)T + (-60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (-5.18 + 13.2i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + (-1.14 + 5.00i)T + (-87.3 - 42.0i)T^{2} \) |
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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
−9.607388674398696588699866543981, −9.033691985564195674566958523551, −8.455002062745089006066192178080, −7.57441376395931655970544036758, −6.66835262241706703310509622585, −5.53521335969353462784628557879, −4.76226508538780045280519547480, −3.14664248649963271450844347275, −1.94499993858947921185550057611, −0.69810026062872681232118391269,
2.43062775601695606715655344070, 3.66662748498383913666446935096, 3.83949161160283296286957638723, 5.33504856984083953951653531963, 6.68543907204655716326904496477, 7.20524712544803945679635521138, 8.282508989896071137123798803125, 9.518464000735638745995368974736, 10.17074877740975154987612018005, 10.49590275841420196614571332273
