L(s) = 1 | + (8.73 + 2.16i)3-s − 14.5·5-s + 70.6·7-s + (71.5 + 37.8i)9-s − 163.·11-s − 293. i·13-s + (−126. − 31.4i)15-s − 533. i·17-s − 314. i·19-s + (617. + 153. i)21-s + 547. i·23-s − 414.·25-s + (543. + 486. i)27-s + 493.·29-s + 1.04e3·31-s + ⋯ |
L(s) = 1 | + (0.970 + 0.240i)3-s − 0.580·5-s + 1.44·7-s + (0.883 + 0.467i)9-s − 1.34·11-s − 1.73i·13-s + (−0.563 − 0.139i)15-s − 1.84i·17-s − 0.870i·19-s + (1.39 + 0.347i)21-s + 1.03i·23-s − 0.663·25-s + (0.745 + 0.666i)27-s + 0.587·29-s + 1.08·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.640325125\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.640325125\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-8.73 - 2.16i)T \) |
good | 5 | \( 1 + 14.5T + 625T^{2} \) |
| 7 | \( 1 - 70.6T + 2.40e3T^{2} \) |
| 11 | \( 1 + 163.T + 1.46e4T^{2} \) |
| 13 | \( 1 + 293. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 533. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 314. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 547. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 493.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.04e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 550. iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 339. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 2.58e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 100. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.40e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 1.43e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 4.99e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 168. iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 5.26e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 1.24e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 7.42e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 116.T + 4.74e7T^{2} \) |
| 89 | \( 1 - 7.44e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 8.27e3T + 8.85e7T^{2} \) |
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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
−10.55967679161068384769089252885, −9.652990873901669483064636382470, −8.419385870751883416864219615768, −7.85454415299457607811659569074, −7.35748210667068393228811902191, −5.22759269939521478017462701720, −4.78781715057890828954716625398, −3.25162551119606568330059051549, −2.38373215486227259431584817804, −0.67086703906433564161267399068,
1.47696418037967423528813079006, 2.37180912297061904723211336175, 3.98833680554137273049515688912, 4.61456035975698843651695737086, 6.20885801559019772750111123820, 7.46805655732300266023791748369, 8.230362940241470114609713374029, 8.551923651968132476362997701762, 9.994485961583311642398127549920, 10.82350948220625370265178114782