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
−12.23415551513354570773242126458, −11.21724286450046764261589734658, −10.54090661468636434887125854461, −9.665314428559700806189066538006, −7.85030492488645205383862848051, −7.08401875754303142180282367380, −6.04782241369654250591952527937, −5.14312765803367505486791020255, −3.75357644742168690915606468563, −2.70013939065451712579105482747,
1.40687266940999992406502657897, 3.09028342377420926082120740966, 4.53922403749489977976799126104, 5.73171875399993510759437981271, 6.27279555187434750159971513059, 7.59973006583486879571076784095, 9.306199513237654962350783161395, 9.725479450750394904398544438208, 11.19245412487668339326036107774, 12.23547056801421923639939982750