| L(s) = 1 | + (0.5 − 1.53i)2-s + (−0.978 − 0.207i)3-s + (−0.5 − 0.363i)4-s + (0.190 + 0.330i)5-s + (−0.809 + 1.40i)6-s + (2.74 − 1.22i)7-s + (1.80 − 1.31i)8-s + (−1.82 − 0.813i)9-s + (0.604 − 0.128i)10-s + (−0.547 + 5.20i)11-s + (0.413 + 0.459i)12-s + (1.24 − 1.37i)13-s + (−0.507 − 4.82i)14-s + (−0.118 − 0.363i)15-s + (−1.50 − 4.61i)16-s + (−0.442 − 4.21i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 1.08i)2-s + (−0.564 − 0.120i)3-s + (−0.250 − 0.181i)4-s + (0.0854 + 0.147i)5-s + (−0.330 + 0.572i)6-s + (1.03 − 0.461i)7-s + (0.639 − 0.464i)8-s + (−0.609 − 0.271i)9-s + (0.191 − 0.0406i)10-s + (−0.165 + 1.57i)11-s + (0.119 + 0.132i)12-s + (0.344 − 0.382i)13-s + (−0.135 − 1.29i)14-s + (−0.0304 − 0.0937i)15-s + (−0.375 − 1.15i)16-s + (−0.107 − 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.293 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.293 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.14399 - 1.54730i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14399 - 1.54730i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 + (-0.5 + 1.53i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (0.978 + 0.207i)T + (2.74 + 1.22i)T^{2} \) |
| 5 | \( 1 + (-0.190 - 0.330i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-2.74 + 1.22i)T + (4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (0.547 - 5.20i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (-1.24 + 1.37i)T + (-1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (0.442 + 4.21i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (-3.34 - 3.71i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-2.80 + 2.04i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.97 + 6.06i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (2.11 - 3.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.41 + 0.513i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-1.59 - 1.77i)T + (-4.49 + 42.7i)T^{2} \) |
| 47 | \( 1 + (1.73 + 5.34i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.646 + 0.288i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (0.516 + 0.109i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + (0.118 + 0.204i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (10.1 + 4.51i)T + (47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (-1.20 + 11.4i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-6.93 + 1.47i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (6.97 + 5.06i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-15.1 - 10.9i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.24168841236530257302247834290, −9.238763404718537141993653691360, −7.935835061891194995717488529955, −7.30926204070863105373838402006, −6.31307001273299068128518568996, −4.97404321959749433326290744373, −4.54178481360563995567953544554, −3.22429529185125859762335658998, −2.18931743034099248309242614530, −0.985790913637980537082113421009,
1.43122146264104049622760965598, 3.04930395988485338611545431367, 4.56826159694958075689126613641, 5.44829372403880284425548401953, 5.69889719848788245889884711545, 6.69243981196566116771974806152, 7.69844919506677295177935177039, 8.556140791193352957149188070400, 8.932216348075982339132801726336, 10.67879907567311849687597866805
