L(s) = 1 | + (1.32 − 0.505i)2-s + (−0.658 − 0.658i)3-s + (1.48 − 1.33i)4-s + (0.820 − 2.07i)5-s + (−1.20 − 0.536i)6-s + (−1.89 + 1.89i)7-s + (1.28 − 2.51i)8-s − 2.13i·9-s + (0.0316 − 3.16i)10-s + 1.63i·11-s + (−1.85 − 0.0997i)12-s + (−0.707 + 0.707i)13-s + (−1.54 + 3.45i)14-s + (−1.90 + 0.828i)15-s + (0.428 − 3.97i)16-s + (2.78 + 2.78i)17-s + ⋯ |
L(s) = 1 | + (0.933 − 0.357i)2-s + (−0.379 − 0.379i)3-s + (0.743 − 0.668i)4-s + (0.367 − 0.930i)5-s + (−0.490 − 0.218i)6-s + (−0.714 + 0.714i)7-s + (0.455 − 0.890i)8-s − 0.711i·9-s + (0.0100 − 0.999i)10-s + 0.493i·11-s + (−0.536 − 0.0287i)12-s + (−0.196 + 0.196i)13-s + (−0.411 + 0.923i)14-s + (−0.492 + 0.213i)15-s + (0.107 − 0.994i)16-s + (0.674 + 0.674i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.123 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.123 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.42966 - 1.26245i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42966 - 1.26245i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.32 + 0.505i)T \) |
| 5 | \( 1 + (-0.820 + 2.07i)T \) |
| 13 | \( 1 + (0.707 - 0.707i)T \) |
good | 3 | \( 1 + (0.658 + 0.658i)T + 3iT^{2} \) |
| 7 | \( 1 + (1.89 - 1.89i)T - 7iT^{2} \) |
| 11 | \( 1 - 1.63iT - 11T^{2} \) |
| 17 | \( 1 + (-2.78 - 2.78i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.52T + 19T^{2} \) |
| 23 | \( 1 + (-0.868 - 0.868i)T + 23iT^{2} \) |
| 29 | \( 1 - 7.97iT - 29T^{2} \) |
| 31 | \( 1 + 7.80iT - 31T^{2} \) |
| 37 | \( 1 + (-4.02 - 4.02i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.49T + 41T^{2} \) |
| 43 | \( 1 + (-5.16 - 5.16i)T + 43iT^{2} \) |
| 47 | \( 1 + (8.12 - 8.12i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.551 - 0.551i)T - 53iT^{2} \) |
| 59 | \( 1 + 8.09T + 59T^{2} \) |
| 61 | \( 1 + 8.43T + 61T^{2} \) |
| 67 | \( 1 + (-1.32 + 1.32i)T - 67iT^{2} \) |
| 71 | \( 1 + 6.50iT - 71T^{2} \) |
| 73 | \( 1 + (4.95 - 4.95i)T - 73iT^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 + (6.57 + 6.57i)T + 83iT^{2} \) |
| 89 | \( 1 + 13.8iT - 89T^{2} \) |
| 97 | \( 1 + (0.0106 + 0.0106i)T + 97iT^{2} \) |
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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
−12.23547056801421923639939982750, −11.19245412487668339326036107774, −9.725479450750394904398544438208, −9.306199513237654962350783161395, −7.59973006583486879571076784095, −6.27279555187434750159971513059, −5.73171875399993510759437981271, −4.53922403749489977976799126104, −3.09028342377420926082120740966, −1.40687266940999992406502657897,
2.70013939065451712579105482747, 3.75357644742168690915606468563, 5.14312765803367505486791020255, 6.04782241369654250591952527937, 7.08401875754303142180282367380, 7.85030492488645205383862848051, 9.665314428559700806189066538006, 10.54090661468636434887125854461, 11.21724286450046764261589734658, 12.23415551513354570773242126458