L(s) = 1 | + (−1.83 + 1.28i)5-s + (−2.12 − 1.57i)7-s + 4.70·11-s + (1.83 + 1.83i)13-s + (0.737 − 0.737i)17-s − 4.75·19-s + (−3.70 + 3.70i)23-s + (1.70 − 4.70i)25-s − 0.701i·29-s + 8.79i·31-s + (5.91 + 0.160i)35-s + (−3.70 − 3.70i)37-s + 4.75i·41-s + (−5 + 5i)43-s + (−8.05 + 8.05i)47-s + ⋯ |
L(s) = 1 | + (−0.818 + 0.574i)5-s + (−0.802 − 0.596i)7-s + 1.41·11-s + (0.507 + 0.507i)13-s + (0.178 − 0.178i)17-s − 1.09·19-s + (−0.771 + 0.771i)23-s + (0.340 − 0.940i)25-s − 0.130i·29-s + 1.58i·31-s + (0.999 + 0.0270i)35-s + (−0.608 − 0.608i)37-s + 0.742i·41-s + (−0.762 + 0.762i)43-s + (−1.17 + 1.17i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7579761179\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7579761179\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.83 - 1.28i)T \) |
| 7 | \( 1 + (2.12 + 1.57i)T \) |
good | 11 | \( 1 - 4.70T + 11T^{2} \) |
| 13 | \( 1 + (-1.83 - 1.83i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.737 + 0.737i)T - 17iT^{2} \) |
| 19 | \( 1 + 4.75T + 19T^{2} \) |
| 23 | \( 1 + (3.70 - 3.70i)T - 23iT^{2} \) |
| 29 | \( 1 + 0.701iT - 29T^{2} \) |
| 31 | \( 1 - 8.79iT - 31T^{2} \) |
| 37 | \( 1 + (3.70 + 3.70i)T + 37iT^{2} \) |
| 41 | \( 1 - 4.75iT - 41T^{2} \) |
| 43 | \( 1 + (5 - 5i)T - 43iT^{2} \) |
| 47 | \( 1 + (8.05 - 8.05i)T - 47iT^{2} \) |
| 53 | \( 1 + (5 - 5i)T - 53iT^{2} \) |
| 59 | \( 1 + 4.75T + 59T^{2} \) |
| 61 | \( 1 - 9.50iT - 61T^{2} \) |
| 67 | \( 1 + (-5 - 5i)T + 67iT^{2} \) |
| 71 | \( 1 - 5.40T + 71T^{2} \) |
| 73 | \( 1 + (-3.11 - 3.11i)T + 73iT^{2} \) |
| 79 | \( 1 + 6.70iT - 79T^{2} \) |
| 83 | \( 1 + (7.86 + 7.86i)T + 83iT^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 + (5.49 - 5.49i)T - 97iT^{2} \) |
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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
−9.962480165409249745451958533136, −9.170261111645590914232022935734, −8.343608749891348852089896740828, −7.37712409450693252349953323905, −6.60652089993961518792538646117, −6.19747587014615194461952458886, −4.53265864365415813784799598020, −3.82544483499963238192736871853, −3.12764958687607544589211564798, −1.44017245707128326641724401355,
0.33198683406876707266111612370, 1.94486014066935690554251263957, 3.48441565108342843905894699797, 4.00133721660409766465527402558, 5.15500762485246576473187794591, 6.26085933905483155632307885946, 6.72797336963340974758556363794, 8.064986852732798656587090692989, 8.552957769365209308519197767686, 9.311083555289766696576910115452