| L(s) = 1 | + (−0.866 + 1.5i)3-s + (−1.73 + 2i)7-s + (−1.5 − 2.59i)9-s − 5.19i·11-s − 1.73·13-s + 3i·17-s + 3.46i·19-s + (−1.50 − 4.33i)21-s + 5.19·27-s + 5.19i·29-s − 10.3i·31-s + (7.79 + 4.5i)33-s − 8i·37-s + (1.49 − 2.59i)39-s + 6·41-s + ⋯ |
| L(s) = 1 | + (−0.499 + 0.866i)3-s + (−0.654 + 0.755i)7-s + (−0.5 − 0.866i)9-s − 1.56i·11-s − 0.480·13-s + 0.727i·17-s + 0.794i·19-s + (−0.327 − 0.944i)21-s + 1.00·27-s + 0.964i·29-s − 1.86i·31-s + (1.35 + 0.783i)33-s − 1.31i·37-s + (0.240 − 0.416i)39-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.000533575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.000533575\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.866 - 1.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.73 - 2i)T \) |
| good | 11 | \( 1 + 5.19iT - 11T^{2} \) |
| 13 | \( 1 + 1.73T + 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 5.19iT - 29T^{2} \) |
| 31 | \( 1 + 10.3iT - 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 10.3iT - 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 - 13T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 1.73T + 97T^{2} \) |
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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
−9.086760051929126549905672604068, −8.608685235329695615898584659150, −7.60288039177269710218721189682, −6.36599078872016862771615166608, −5.83015419643326720155934278206, −5.34713168342070028744374548131, −4.01651711188077520853404974865, −3.45986895431923281011080974874, −2.38576024826760660452815000002, −0.50050537122063799043227227567,
0.879099107620023522213250642975, 2.14182174511924910098334526443, 3.08526204860827565699904950972, 4.53683624726191158260482294462, 4.97562983588756588615333379080, 6.22329691184891045806593588182, 6.94291828178758684615854179697, 7.28685009117967465440684900857, 8.094642566147838892494360305446, 9.252037355509663039321098724309
