L(s) = 1 | − 0.414·2-s + (1 − 1.41i)3-s − 1.82·4-s + i·5-s + (−0.414 + 0.585i)6-s − i·7-s + 1.58·8-s + (−1.00 − 2.82i)9-s − 0.414i·10-s + (−1.41 + 3i)11-s + (−1.82 + 2.58i)12-s + 0.414i·14-s + (1.41 + i)15-s + 3·16-s − 6.82·17-s + (0.414 + 1.17i)18-s + ⋯ |
L(s) = 1 | − 0.292·2-s + (0.577 − 0.816i)3-s − 0.914·4-s + 0.447i·5-s + (−0.169 + 0.239i)6-s − 0.377i·7-s + 0.560·8-s + (−0.333 − 0.942i)9-s − 0.130i·10-s + (−0.426 + 0.904i)11-s + (−0.527 + 0.746i)12-s + 0.110i·14-s + (0.365 + 0.258i)15-s + 0.750·16-s − 1.65·17-s + (0.0976 + 0.276i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2860929498\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2860929498\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1 + 1.41i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (1.41 - 3i)T \) |
good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 6.82T + 17T^{2} \) |
| 19 | \( 1 - 0.828iT - 19T^{2} \) |
| 23 | \( 1 - 0.828iT - 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 - 0.828T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 7.65T + 41T^{2} \) |
| 43 | \( 1 - 9.65iT - 43T^{2} \) |
| 47 | \( 1 - 10.8iT - 47T^{2} \) |
| 53 | \( 1 - 0.828iT - 53T^{2} \) |
| 59 | \( 1 + 0.343iT - 59T^{2} \) |
| 61 | \( 1 + 5.17iT - 61T^{2} \) |
| 67 | \( 1 + 6.48T + 67T^{2} \) |
| 71 | \( 1 - 0.343iT - 71T^{2} \) |
| 73 | \( 1 - 12iT - 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 7.65iT - 89T^{2} \) |
| 97 | \( 1 + 5.17T + 97T^{2} \) |
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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
−9.763403006653001658408920812914, −9.289856965248178960035514613006, −8.344237512109027525599071546248, −7.67794365889803871743854154332, −6.98849592511675677782244669499, −6.06939081900664290088678629777, −4.74958622568013601445870104712, −3.94154965762561708019370821348, −2.70989985917337703762010368529, −1.55221460805115784240836992089,
0.12809109501239173873757367167, 2.12058639957554294688733166217, 3.40730643943839990502081614961, 4.31165667970567572455813976208, 5.06478272046453998213457495403, 5.85660008692696013102843528862, 7.31656896123235513198865958480, 8.466839961494582344750488637517, 8.637552329127722349039570511866, 9.325903554682294466246927223243