| L(s) = 1 | + (0.282 + 1.60i)2-s + (−0.824 + 0.691i)3-s + (−0.607 + 0.221i)4-s + (0.0661 + 0.0240i)5-s + (−1.34 − 1.12i)6-s + (1.10 + 1.90i)8-s + (−0.319 + 1.81i)9-s + (−0.0198 + 0.112i)10-s + (−0.895 − 1.55i)11-s + (0.347 − 0.602i)12-s + (2.27 + 1.90i)13-s + (−0.0711 + 0.0258i)15-s + (−3.73 + 3.13i)16-s + (0.109 + 0.620i)17-s − 2.99·18-s + (3.31 + 2.82i)19-s + ⋯ |
| L(s) = 1 | + (0.199 + 1.13i)2-s + (−0.475 + 0.399i)3-s + (−0.303 + 0.110i)4-s + (0.0295 + 0.0107i)5-s + (−0.547 − 0.459i)6-s + (0.389 + 0.674i)8-s + (−0.106 + 0.604i)9-s + (−0.00628 + 0.0356i)10-s + (−0.270 − 0.467i)11-s + (0.100 − 0.173i)12-s + (0.630 + 0.529i)13-s + (−0.0183 + 0.00668i)15-s + (−0.933 + 0.783i)16-s + (0.0265 + 0.150i)17-s − 0.706·18-s + (0.760 + 0.649i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00537175 - 1.40843i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00537175 - 1.40843i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 19 | \( 1 + (-3.31 - 2.82i)T \) |
| good | 2 | \( 1 + (-0.282 - 1.60i)T + (-1.87 + 0.684i)T^{2} \) |
| 3 | \( 1 + (0.824 - 0.691i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (-0.0661 - 0.0240i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (0.895 + 1.55i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.27 - 1.90i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.109 - 0.620i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-1.40 + 0.513i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.942 - 5.34i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (3.26 - 5.65i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.691T + 37T^{2} \) |
| 41 | \( 1 + (5.15 - 4.32i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (1.77 + 0.645i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.241 + 1.36i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (6.93 - 2.52i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (2.40 + 13.6i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (0.157 - 0.0574i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.60 - 9.09i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-11.0 - 4.03i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-8.59 + 7.21i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-8.02 + 6.73i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (3.37 - 5.85i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.16 - 2.65i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-0.928 - 5.26i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.64520629402807326524305346318, −9.609645343511138188781823785574, −8.468246092158827409286486158617, −7.944416858991456587510041409978, −6.93778589308491238838137442463, −6.13063109600439195301024126544, −5.38537313090151651345273364782, −4.72643740995223557999232287671, −3.45202506242458902098945204011, −1.83192278002663341615829213465,
0.65680460591724820128343047426, 1.89360500429755114807754804084, 3.11342313752810653922121536306, 3.93832260482454863788434497096, 5.18376489610380547924409481465, 6.13831576409143131462135969901, 7.09975585739132183712191732088, 7.82904076289143451947791049523, 9.228044788671095777157790184363, 9.750851586860816678243207909865
