| L(s) = 1 | + (−1.34 − 0.437i)2-s + (−2.31 − 2.08i)3-s + (0.00296 + 0.00215i)4-s + (2.20 + 3.82i)6-s + (−0.944 − 0.0992i)7-s + (1.66 + 2.28i)8-s + (0.702 + 6.68i)9-s + (2.64 − 1.17i)11-s + (−0.00237 − 0.0111i)12-s + (0.844 − 3.97i)13-s + (1.22 + 0.546i)14-s + (−1.23 − 3.81i)16-s + (2.40 − 5.39i)17-s + (1.97 − 9.31i)18-s + (3.85 − 0.819i)19-s + ⋯ |
| L(s) = 1 | + (−0.951 − 0.309i)2-s + (−1.33 − 1.20i)3-s + (0.00148 + 0.00107i)4-s + (0.901 + 1.56i)6-s + (−0.357 − 0.0375i)7-s + (0.587 + 0.808i)8-s + (0.234 + 2.22i)9-s + (0.798 − 0.355i)11-s + (−0.000685 − 0.00322i)12-s + (0.234 − 1.10i)13-s + (0.328 + 0.146i)14-s + (−0.309 − 0.952i)16-s + (0.582 − 1.30i)17-s + (0.466 − 2.19i)18-s + (0.884 − 0.187i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 + 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.127966 - 0.494934i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.127966 - 0.494934i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 + (-4.68 + 3.01i)T \) |
| good | 2 | \( 1 + (1.34 + 0.437i)T + (1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (2.31 + 2.08i)T + (0.313 + 2.98i)T^{2} \) |
| 7 | \( 1 + (0.944 + 0.0992i)T + (6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (-2.64 + 1.17i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (-0.844 + 3.97i)T + (-11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (-2.40 + 5.39i)T + (-11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-3.85 + 0.819i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-4.17 - 5.74i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.690 + 2.12i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (1.26 - 0.728i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.928 - 1.03i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-2.66 - 12.5i)T + (-39.2 + 17.4i)T^{2} \) |
| 47 | \( 1 + (-1.42 + 0.462i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (2.69 - 0.282i)T + (51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-2.76 + 3.07i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 5.75T + 61T^{2} \) |
| 67 | \( 1 + (-7.95 - 4.59i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.16 + 11.0i)T + (-69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (2.19 + 4.92i)T + (-48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (3.76 + 1.67i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-12.5 + 11.3i)T + (8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (6.19 + 4.50i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (0.00200 - 0.00275i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−9.923462728110679972712856965872, −9.322878929017480967638055624663, −8.003794709749489751005013674401, −7.51548576708251076729143700376, −6.52683990579737899567970413547, −5.60414900193343084889175566856, −4.93583058996074071358626753494, −2.95672762772210291677348844014, −1.29439112833931558686865496572, −0.62003836085474551899562732701,
1.09658761541219789167360681598, 3.69571203629130588345427572389, 4.28869608525626797115253870759, 5.35127643299014266001739728761, 6.49895429778204018011421352384, 6.92661815362032932728951786124, 8.449960755370393880005403075322, 9.160744822256403461894127865388, 9.832671758583523335249961070786, 10.42713201899063627943628248024
