| L(s) = 1 | + (−0.809 + 0.587i)2-s + (0.913 − 0.406i)3-s + (0.309 − 0.951i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.348 − 0.386i)7-s + (0.309 + 0.951i)8-s + (0.669 − 0.743i)9-s + (0.913 + 0.406i)10-s + (5.88 − 1.25i)11-s + (−0.104 − 0.994i)12-s + (0.258 − 2.46i)13-s + (0.509 + 0.108i)14-s + (−0.809 − 0.587i)15-s + (−0.809 − 0.587i)16-s + (−7.02 − 1.49i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.527 − 0.234i)3-s + (0.154 − 0.475i)4-s + (−0.223 − 0.387i)5-s + (−0.204 + 0.353i)6-s + (−0.131 − 0.146i)7-s + (0.109 + 0.336i)8-s + (0.223 − 0.247i)9-s + (0.288 + 0.128i)10-s + (1.77 − 0.377i)11-s + (−0.0301 − 0.287i)12-s + (0.0717 − 0.682i)13-s + (0.136 + 0.0289i)14-s + (−0.208 − 0.151i)15-s + (−0.202 − 0.146i)16-s + (−1.70 − 0.361i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.449 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.10539 - 0.681306i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10539 - 0.681306i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 3 | \( 1 + (-0.913 + 0.406i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (-5.30 + 1.69i)T \) |
| good | 7 | \( 1 + (0.348 + 0.386i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (-5.88 + 1.25i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.258 + 2.46i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (7.02 + 1.49i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (0.0465 + 0.443i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (0.199 + 0.613i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (6.09 - 4.43i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (-4.31 + 7.47i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.12 - 0.945i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (0.994 + 9.46i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (-4.39 - 3.19i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (2.03 - 2.25i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (-1.45 + 0.648i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + 2.82T + 61T^{2} \) |
| 67 | \( 1 + (0.993 + 1.72i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.52 + 5.02i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (-9.08 + 1.93i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (7.14 + 1.51i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (1.51 + 0.672i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (-1.51 + 4.66i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-2.39 + 7.35i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−9.485262423013924789426280670102, −9.028369366099167771027349170885, −8.427070384148594275428081849317, −7.38139854738650247813749249935, −6.70825896492408581915421386272, −5.84944743525587582888138262936, −4.50335973809215307350735868287, −3.59354051216628739453960497936, −2.10034425433229295322081339496, −0.72933586665305517598234925999,
1.57188402843633888939450333642, 2.64573845226045579507528209057, 3.93439321569881395382125150362, 4.40256473670191515614969284078, 6.33047680132353635383164984946, 6.79524495448176127033955339092, 7.88697721982385405513453776850, 8.796460586139093385575290117937, 9.351374526735878257698387694860, 9.978306372662643787609033087992
