L(s) = 1 | + (0.707 + 0.707i)2-s + (−0.349 − 1.69i)3-s + 1.00i·4-s + (0.707 − 0.707i)5-s + (0.952 − 1.44i)6-s + 2.97·7-s + (−0.707 + 0.707i)8-s + (−2.75 + 1.18i)9-s + 1.00·10-s − 2.43·11-s + (1.69 − 0.349i)12-s + (3.37 + 3.37i)13-s + (2.10 + 2.10i)14-s + (−1.44 − 0.952i)15-s − 1.00·16-s + (2.64 − 2.64i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (−0.201 − 0.979i)3-s + 0.500i·4-s + (0.316 − 0.316i)5-s + (0.388 − 0.590i)6-s + 1.12·7-s + (−0.250 + 0.250i)8-s + (−0.918 + 0.395i)9-s + 0.316·10-s − 0.735·11-s + (0.489 − 0.100i)12-s + (0.937 + 0.937i)13-s + (0.562 + 0.562i)14-s + (−0.373 − 0.245i)15-s − 0.250·16-s + (0.640 − 0.640i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.111i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.372079594\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.372079594\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (0.349 + 1.69i)T \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
| 37 | \( 1 + (-5.17 + 3.19i)T \) |
good | 7 | \( 1 - 2.97T + 7T^{2} \) |
| 11 | \( 1 + 2.43T + 11T^{2} \) |
| 13 | \( 1 + (-3.37 - 3.37i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.64 + 2.64i)T - 17iT^{2} \) |
| 19 | \( 1 + (-5.18 - 5.18i)T + 19iT^{2} \) |
| 23 | \( 1 + (-4.56 + 4.56i)T - 23iT^{2} \) |
| 29 | \( 1 + (2.17 + 2.17i)T + 29iT^{2} \) |
| 31 | \( 1 + (1.20 - 1.20i)T - 31iT^{2} \) |
| 41 | \( 1 + 7.04T + 41T^{2} \) |
| 43 | \( 1 + (3.83 + 3.83i)T + 43iT^{2} \) |
| 47 | \( 1 + 6.56iT - 47T^{2} \) |
| 53 | \( 1 + 0.00856iT - 53T^{2} \) |
| 59 | \( 1 + (-3.57 + 3.57i)T - 59iT^{2} \) |
| 61 | \( 1 + (-8.28 + 8.28i)T - 61iT^{2} \) |
| 67 | \( 1 + 3.64iT - 67T^{2} \) |
| 71 | \( 1 - 9.63iT - 71T^{2} \) |
| 73 | \( 1 - 5.75iT - 73T^{2} \) |
| 79 | \( 1 + (-6.34 - 6.34i)T + 79iT^{2} \) |
| 83 | \( 1 - 16.7iT - 83T^{2} \) |
| 89 | \( 1 + (-1.72 - 1.72i)T + 89iT^{2} \) |
| 97 | \( 1 + (4.30 + 4.30i)T + 97iT^{2} \) |
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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
−9.728201815845194563939638810110, −8.554733301819541868683051098829, −8.117499939136306126339669481137, −7.28929799064365432447424604635, −6.49382886611279238107988175079, −5.38970128882798277105219433424, −5.12278776556729894737674461856, −3.68763574246663543830229668010, −2.31042709735400199509680647839, −1.20661509247155383175520678071,
1.24990556964911917802379451008, 2.87292903871650295924238537800, 3.51824914703894107216010870768, 4.83403045770655277600183966846, 5.29816578588310987533648911783, 6.04390126872269886363025170689, 7.45317320116182163999527294293, 8.336964915384092175099964931809, 9.261286987591339957630662492048, 10.09291470181718321690494610462