| L(s) = 1 | + (−0.0747 − 0.997i)2-s + (−1.51 + 0.838i)3-s + (−0.988 + 0.149i)4-s + (−0.613 + 0.418i)5-s + (0.949 + 1.44i)6-s + (0.986 + 0.148i)7-s + (0.222 + 0.974i)8-s + (1.59 − 2.54i)9-s + (0.463 + 0.580i)10-s + (−0.221 + 0.0682i)11-s + (1.37 − 1.05i)12-s + (0.329 + 0.305i)13-s + (0.0745 − 0.995i)14-s + (0.579 − 1.14i)15-s + (0.955 − 0.294i)16-s − 7.63·17-s + ⋯ |
| L(s) = 1 | + (−0.0528 − 0.705i)2-s + (−0.875 + 0.484i)3-s + (−0.494 + 0.0745i)4-s + (−0.274 + 0.187i)5-s + (0.387 + 0.591i)6-s + (0.372 + 0.0562i)7-s + (0.0786 + 0.344i)8-s + (0.531 − 0.847i)9-s + (0.146 + 0.183i)10-s + (−0.0666 + 0.0205i)11-s + (0.396 − 0.304i)12-s + (0.0913 + 0.0848i)13-s + (0.0199 − 0.265i)14-s + (0.149 − 0.296i)15-s + (0.238 − 0.0736i)16-s − 1.85·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0441902 + 0.126191i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0441902 + 0.126191i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0747 + 0.997i)T \) |
| 3 | \( 1 + (1.51 - 0.838i)T \) |
| 29 | \( 1 + (5.26 - 1.14i)T \) |
| good | 5 | \( 1 + (0.613 - 0.418i)T + (1.82 - 4.65i)T^{2} \) |
| 7 | \( 1 + (-0.986 - 0.148i)T + (6.68 + 2.06i)T^{2} \) |
| 11 | \( 1 + (0.221 - 0.0682i)T + (9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (-0.329 - 0.305i)T + (0.971 + 12.9i)T^{2} \) |
| 17 | \( 1 + 7.63T + 17T^{2} \) |
| 19 | \( 1 + (4.08 + 5.12i)T + (-4.22 + 18.5i)T^{2} \) |
| 23 | \( 1 + (0.617 - 8.24i)T + (-22.7 - 3.42i)T^{2} \) |
| 31 | \( 1 + (2.12 - 1.45i)T + (11.3 - 28.8i)T^{2} \) |
| 37 | \( 1 + (0.995 + 4.36i)T + (-33.3 + 16.0i)T^{2} \) |
| 41 | \( 1 + (-4.77 + 8.26i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.19 - 2.17i)T + (15.7 + 40.0i)T^{2} \) |
| 47 | \( 1 + (6.96 - 2.14i)T + (38.8 - 26.4i)T^{2} \) |
| 53 | \( 1 + (12.0 + 5.79i)T + (33.0 + 41.4i)T^{2} \) |
| 59 | \( 1 + (6.99 - 12.1i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.09 + 0.164i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (-10.5 - 3.26i)T + (55.3 + 37.7i)T^{2} \) |
| 71 | \( 1 + (0.802 - 3.51i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-8.33 + 4.01i)T + (45.5 - 57.0i)T^{2} \) |
| 79 | \( 1 + (-3.79 + 3.51i)T + (5.90 - 78.7i)T^{2} \) |
| 83 | \( 1 + (-2.72 + 6.95i)T + (-60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (-5.55 - 2.67i)T + (55.4 + 69.5i)T^{2} \) |
| 97 | \( 1 + (-0.0468 + 0.119i)T + (-71.1 - 65.9i)T^{2} \) |
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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
−11.07703722250250685521768004466, −10.82034210220314417265183480703, −9.408674990843209053007272200926, −9.021151420450744016961784345538, −7.57555890580107253740399505788, −6.58559907276164278617419167229, −5.39468668072339319495953001243, −4.49805592498400108512359583665, −3.58011224795478017015460597380, −1.91492251113219216975977391196,
0.085964495435692189981341044055, 2.00256667117142779229562431595, 4.23115842280522340233571007879, 4.85556281052006183113843973501, 6.24995072274866954580317967913, 6.53660516264485563381801891511, 7.918568997470957331333042092756, 8.313143153772402527595304548073, 9.598678634128498431928051746939, 10.75303244383970866465489166002
