L(s) = 1 | + (1.95 − 4.81i)3-s + 5i·5-s + 20.3i·7-s + (−19.3 − 18.8i)9-s + 15.2·11-s − 27.7·13-s + (24.0 + 9.77i)15-s + 92.5i·17-s − 127. i·19-s + (98.0 + 39.8i)21-s + 51.1·23-s − 25·25-s + (−128. + 56.3i)27-s − 99.2i·29-s + 25.8i·31-s + ⋯ |
L(s) = 1 | + (0.376 − 0.926i)3-s + 0.447i·5-s + 1.09i·7-s + (−0.716 − 0.697i)9-s + 0.419·11-s − 0.591·13-s + (0.414 + 0.168i)15-s + 1.31i·17-s − 1.54i·19-s + (1.01 + 0.413i)21-s + 0.463·23-s − 0.200·25-s + (−0.915 + 0.401i)27-s − 0.635i·29-s + 0.149i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.376i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.926 - 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.1051659160\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1051659160\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.95 + 4.81i)T \) |
| 5 | \( 1 - 5iT \) |
good | 7 | \( 1 - 20.3iT - 343T^{2} \) |
| 11 | \( 1 - 15.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 27.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 92.5iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 127. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 51.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 99.2iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 25.8iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 356.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 292. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 521. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 573.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 305. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 295.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 326.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 299. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 653.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 504.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 110. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 1.47e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 772. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 26.1T + 9.12e5T^{2} \) |
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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
−9.834941356007534520887399524720, −8.960755933816433738632011685622, −8.433547118453732263662505896876, −7.42856898167080592230913232517, −6.62565940044158337399702722507, −5.97038102279118086448742840998, −4.86029141691017624835652815767, −3.39251035690016192900907715580, −2.53083077727456322366008614527, −1.61827679633992664991940625838,
0.02362975562704316607174619754, 1.51839245666203305436447804590, 3.05015850775215286093019723529, 3.91633687469922988168743649699, 4.74823920291499836983868449334, 5.51402294747150626335802456186, 6.88573354744927603317952532338, 7.66558171699736793970366882020, 8.563209522060122935851808738794, 9.397870055736578215492818795726