L(s) = 1 | + (−5.09 + i)3-s − 10.1·5-s + 10.1i·7-s + (24.9 − 10.1i)9-s + 46i·11-s + 44i·13-s + (51.9 − 10.1i)15-s − 20.3i·17-s + 10.1·19-s + (−10.1 − 51.9i)21-s + 88·23-s − 21.0·25-s + (−117. + 76.9i)27-s − 254.·29-s + 214. i·31-s + ⋯ |
L(s) = 1 | + (−0.981 + 0.192i)3-s − 0.912·5-s + 0.550i·7-s + (0.925 − 0.377i)9-s + 1.26i·11-s + 0.938i·13-s + (0.895 − 0.175i)15-s − 0.290i·17-s + 0.123·19-s + (−0.105 − 0.540i)21-s + 0.797·23-s − 0.168·25-s + (−0.835 + 0.548i)27-s − 1.63·29-s + 1.24i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.557 + 0.829i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.557 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.03208390264\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03208390264\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (5.09 - i)T \) |
good | 5 | \( 1 + 10.1T + 125T^{2} \) |
| 7 | \( 1 - 10.1iT - 343T^{2} \) |
| 11 | \( 1 - 46iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 44iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 20.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 10.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 88T + 1.21e4T^{2} \) |
| 29 | \( 1 + 254.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 214. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 332iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 489. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 234.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 384T + 1.03e5T^{2} \) |
| 53 | \( 1 + 458.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 630iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 236iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 50.9T + 3.00e5T^{2} \) |
| 71 | \( 1 + 680T + 3.57e5T^{2} \) |
| 73 | \( 1 - 422T + 3.89e5T^{2} \) |
| 79 | \( 1 - 744. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 186iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 958. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.06e3T + 9.12e5T^{2} \) |
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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
−10.87024522153059107571538305943, −9.670632328835209026584899062946, −8.963728605218476695593593723919, −7.43854969840887024066672045858, −6.96451130113298129288319355063, −5.62282530966299137416364506657, −4.67686447436466987560723384350, −3.76004204407560799852231956181, −1.85910062992479554928979402842, −0.01506260551987765355658224325,
1.06355969408300547519815520108, 3.25253909034725465449400694267, 4.32133106998167298688773328805, 5.51095416516268299849557402167, 6.38999803452639275995949366759, 7.59686438100540575256188407735, 8.098748868606402033578624049840, 9.555650939802339052621897608545, 10.67370075703918632218959724529, 11.23310887990513646887586348840