L(s) = 1 | + 4.29i·5-s + i·7-s − 4.77·11-s − 0.0836·13-s + 2.60i·17-s + 4.85i·19-s + 4.78·23-s − 13.4·25-s − 0.263i·29-s + 0.304i·31-s − 4.29·35-s + 9.56·37-s + 10.5i·41-s − 4.24i·43-s − 9.85·47-s + ⋯ |
L(s) = 1 | + 1.92i·5-s + 0.377i·7-s − 1.43·11-s − 0.0231·13-s + 0.632i·17-s + 1.11i·19-s + 0.996·23-s − 2.69·25-s − 0.0488i·29-s + 0.0546i·31-s − 0.726·35-s + 1.57·37-s + 1.65i·41-s − 0.647i·43-s − 1.43·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8549924311\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8549924311\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 - 4.29iT - 5T^{2} \) |
| 11 | \( 1 + 4.77T + 11T^{2} \) |
| 13 | \( 1 + 0.0836T + 13T^{2} \) |
| 17 | \( 1 - 2.60iT - 17T^{2} \) |
| 19 | \( 1 - 4.85iT - 19T^{2} \) |
| 23 | \( 1 - 4.78T + 23T^{2} \) |
| 29 | \( 1 + 0.263iT - 29T^{2} \) |
| 31 | \( 1 - 0.304iT - 31T^{2} \) |
| 37 | \( 1 - 9.56T + 37T^{2} \) |
| 41 | \( 1 - 10.5iT - 41T^{2} \) |
| 43 | \( 1 + 4.24iT - 43T^{2} \) |
| 47 | \( 1 + 9.85T + 47T^{2} \) |
| 53 | \( 1 - 11.9iT - 53T^{2} \) |
| 59 | \( 1 - 5.50T + 59T^{2} \) |
| 61 | \( 1 + 8.62T + 61T^{2} \) |
| 67 | \( 1 - 1.80iT - 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 4.88T + 73T^{2} \) |
| 79 | \( 1 + 14.3iT - 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 7.85iT - 89T^{2} \) |
| 97 | \( 1 + 1.29T + 97T^{2} \) |
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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
−8.196818610240187970186856889753, −7.76896572091914993426998675080, −7.18903248212038713479341764241, −6.21974740075884577652106537651, −5.99508031646012975667187875609, −4.99592114848699685556704752222, −3.98596594091835624998261590244, −2.96957178842509979737541940422, −2.78280029329075592525691266367, −1.70250136152848979358903952546,
0.24310786347430728353857114589, 0.961752697557961792999523556966, 2.12874180197534926444491311027, 3.07442816261977578157041197100, 4.21176219993687116799335647940, 4.98749897418931503585900724647, 5.09430248949306407229840007823, 6.03620828287775516642692144026, 7.12972727522072825815405020839, 7.72523387124255530758134327087