| L(s) = 1 | + (−1.79 − 0.0724i)2-s + (−0.278 + 0.960i)3-s + (1.22 + 0.0992i)4-s + (1.73 + 0.658i)5-s + (0.569 − 1.70i)6-s + (1.36 + 3.19i)7-s + (1.36 + 0.166i)8-s + (−0.845 − 0.534i)9-s + (−3.07 − 1.30i)10-s + (2.46 + 3.90i)11-s + (−0.437 + 1.15i)12-s + (−3.37 − 1.26i)13-s + (−2.21 − 5.84i)14-s + (−1.11 + 1.48i)15-s + (−4.88 − 0.793i)16-s + (4.77 − 2.03i)17-s + ⋯ |
| L(s) = 1 | + (−1.27 − 0.0511i)2-s + (−0.160 + 0.554i)3-s + (0.614 + 0.0496i)4-s + (0.776 + 0.294i)5-s + (0.232 − 0.696i)6-s + (0.514 + 1.20i)7-s + (0.484 + 0.0587i)8-s + (−0.281 − 0.178i)9-s + (−0.971 − 0.414i)10-s + (0.744 + 1.17i)11-s + (−0.126 + 0.332i)12-s + (−0.936 − 0.349i)13-s + (−0.592 − 1.56i)14-s + (−0.288 + 0.383i)15-s + (−1.22 − 0.198i)16-s + (1.15 − 0.493i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.106 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.546012 + 0.607648i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.546012 + 0.607648i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.278 - 0.960i)T \) |
| 13 | \( 1 + (3.37 + 1.26i)T \) |
| good | 2 | \( 1 + (1.79 + 0.0724i)T + (1.99 + 0.160i)T^{2} \) |
| 5 | \( 1 + (-1.73 - 0.658i)T + (3.74 + 3.31i)T^{2} \) |
| 7 | \( 1 + (-1.36 - 3.19i)T + (-4.84 + 5.04i)T^{2} \) |
| 11 | \( 1 + (-2.46 - 3.90i)T + (-4.71 + 9.93i)T^{2} \) |
| 17 | \( 1 + (-4.77 + 2.03i)T + (11.7 - 12.2i)T^{2} \) |
| 19 | \( 1 + (-4.15 + 2.39i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.442 + 0.765i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.225 - 5.59i)T + (-28.9 - 2.33i)T^{2} \) |
| 31 | \( 1 + (-2.03 - 2.29i)T + (-3.73 + 30.7i)T^{2} \) |
| 37 | \( 1 + (10.1 - 2.07i)T + (34.0 - 14.5i)T^{2} \) |
| 41 | \( 1 + (-4.35 - 1.26i)T + (34.6 + 21.9i)T^{2} \) |
| 43 | \( 1 + (0.221 - 1.08i)T + (-39.5 - 16.8i)T^{2} \) |
| 47 | \( 1 + (2.51 - 1.73i)T + (16.6 - 43.9i)T^{2} \) |
| 53 | \( 1 + (1.00 - 8.26i)T + (-51.4 - 12.6i)T^{2} \) |
| 59 | \( 1 + (1.55 + 9.56i)T + (-55.9 + 18.6i)T^{2} \) |
| 61 | \( 1 + (5.01 - 3.76i)T + (16.9 - 58.5i)T^{2} \) |
| 67 | \( 1 + (-0.331 - 4.10i)T + (-66.1 + 10.7i)T^{2} \) |
| 71 | \( 1 + (-0.0431 + 0.0414i)T + (2.85 - 70.9i)T^{2} \) |
| 73 | \( 1 + (-1.09 + 2.09i)T + (-41.4 - 60.0i)T^{2} \) |
| 79 | \( 1 + (-5.17 - 7.50i)T + (-28.0 + 73.8i)T^{2} \) |
| 83 | \( 1 + (-0.824 + 3.34i)T + (-73.4 - 38.5i)T^{2} \) |
| 89 | \( 1 + (-9.75 - 5.63i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.24 + 5.91i)T + (19.4 + 95.0i)T^{2} \) |
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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
−10.79427108929195687306838810560, −9.872213475774794434898914121556, −9.554311818510924255409906787609, −8.815595001046374653007373808323, −7.71558123128376866453824933296, −6.82715392881737174773250513419, −5.42472191040714059571380882888, −4.78603637682016652080805085600, −2.81356994015381122603987659246, −1.61730912496908131813607763193,
0.846589412815260955435460005198, 1.74513432545008404875699726997, 3.75931291995397169472827980040, 5.19953908382337673306336326232, 6.29302438371682457713129559841, 7.43827166129108511768757775488, 7.88257065493672071795298366307, 8.936655401815306429139280011061, 9.781827406211320767397309645668, 10.38743069213675330460160055817
