| L(s) = 1 | + (186. − 186. i)3-s + (−473. − 3.08e3i)5-s + (−7.25e3 − 7.25e3i)7-s − 1.04e4i·9-s − 3.22e4·11-s + (−1.23e5 + 1.23e5i)13-s + (−6.63e5 − 4.87e5i)15-s + (−1.34e6 − 1.34e6i)17-s − 3.80e6i·19-s − 2.70e6·21-s + (−1.48e6 + 1.48e6i)23-s + (−9.31e6 + 2.92e6i)25-s + (9.06e6 + 9.06e6i)27-s − 1.97e7i·29-s + 3.84e7·31-s + ⋯ |
| L(s) = 1 | + (0.766 − 0.766i)3-s + (−0.151 − 0.988i)5-s + (−0.431 − 0.431i)7-s − 0.176i·9-s − 0.200·11-s + (−0.332 + 0.332i)13-s + (−0.874 − 0.641i)15-s + (−0.947 − 0.947i)17-s − 1.53i·19-s − 0.661·21-s + (−0.230 + 0.230i)23-s + (−0.954 + 0.299i)25-s + (0.631 + 0.631i)27-s − 0.963i·29-s + 1.34·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.568141 - 1.54266i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.568141 - 1.54266i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (473. + 3.08e3i)T \) |
| good | 3 | \( 1 + (-186. + 186. i)T - 5.90e4iT^{2} \) |
| 7 | \( 1 + (7.25e3 + 7.25e3i)T + 2.82e8iT^{2} \) |
| 11 | \( 1 + 3.22e4T + 2.59e10T^{2} \) |
| 13 | \( 1 + (1.23e5 - 1.23e5i)T - 1.37e11iT^{2} \) |
| 17 | \( 1 + (1.34e6 + 1.34e6i)T + 2.01e12iT^{2} \) |
| 19 | \( 1 + 3.80e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 + (1.48e6 - 1.48e6i)T - 4.14e13iT^{2} \) |
| 29 | \( 1 + 1.97e7iT - 4.20e14T^{2} \) |
| 31 | \( 1 - 3.84e7T + 8.19e14T^{2} \) |
| 37 | \( 1 + (-6.70e7 - 6.70e7i)T + 4.80e15iT^{2} \) |
| 41 | \( 1 - 1.28e8T + 1.34e16T^{2} \) |
| 43 | \( 1 + (3.08e7 - 3.08e7i)T - 2.16e16iT^{2} \) |
| 47 | \( 1 + (1.99e8 + 1.99e8i)T + 5.25e16iT^{2} \) |
| 53 | \( 1 + (-3.65e8 + 3.65e8i)T - 1.74e17iT^{2} \) |
| 59 | \( 1 - 1.07e9iT - 5.11e17T^{2} \) |
| 61 | \( 1 - 2.62e8T + 7.13e17T^{2} \) |
| 67 | \( 1 + (5.30e8 + 5.30e8i)T + 1.82e18iT^{2} \) |
| 71 | \( 1 - 2.34e9T + 3.25e18T^{2} \) |
| 73 | \( 1 + (-1.92e9 + 1.92e9i)T - 4.29e18iT^{2} \) |
| 79 | \( 1 - 3.63e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 + (1.23e9 - 1.23e9i)T - 1.55e19iT^{2} \) |
| 89 | \( 1 + 9.01e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + (9.05e9 + 9.05e9i)T + 7.37e19iT^{2} \) |
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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
−15.46510987509701661614812206496, −13.62674032840443117764120182900, −13.18480260281193080531088187241, −11.62478024943763917257818629510, −9.529525612942780174511638082498, −8.273176509770167663158161169929, −6.94339302973738197276587987610, −4.65185746941958984072164437948, −2.46832017581561429175349480866, −0.63038750875085807939759737741,
2.62078721352958959184758762899, 3.92077118920240471857004232328, 6.25358883697008138147833508289, 8.126329989304146389859349125206, 9.633085400940796456360660367645, 10.71256222620951578625906871920, 12.49324184935994743625217417394, 14.25203975904716243440921395883, 15.06831356766698352667502352953, 16.01586383735841846832565774060
