L(s) = 1 | + (1.39 + 0.238i)2-s + (0.183 + 0.183i)3-s + (1.88 + 0.666i)4-s + (−2.16 − 0.569i)5-s + (0.212 + 0.300i)6-s − 3.84·7-s + (2.46 + 1.37i)8-s − 2.93i·9-s + (−2.87 − 1.31i)10-s + (1.60 + 1.60i)11-s + (0.224 + 0.469i)12-s + (1.80 + 1.80i)13-s + (−5.36 − 0.919i)14-s + (−0.292 − 0.502i)15-s + (3.11 + 2.51i)16-s − 4.93i·17-s + ⋯ |
L(s) = 1 | + (0.985 + 0.168i)2-s + (0.106 + 0.106i)3-s + (0.942 + 0.333i)4-s + (−0.966 − 0.254i)5-s + (0.0866 + 0.122i)6-s − 1.45·7-s + (0.873 + 0.487i)8-s − 0.977i·9-s + (−0.910 − 0.414i)10-s + (0.482 + 0.482i)11-s + (0.0647 + 0.135i)12-s + (0.501 + 0.501i)13-s + (−1.43 − 0.245i)14-s + (−0.0755 − 0.129i)15-s + (0.778 + 0.628i)16-s − 1.19i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.238i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36410 + 0.165407i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36410 + 0.165407i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.39 - 0.238i)T \) |
| 5 | \( 1 + (2.16 + 0.569i)T \) |
good | 3 | \( 1 + (-0.183 - 0.183i)T + 3iT^{2} \) |
| 7 | \( 1 + 3.84T + 7T^{2} \) |
| 11 | \( 1 + (-1.60 - 1.60i)T + 11iT^{2} \) |
| 13 | \( 1 + (-1.80 - 1.80i)T + 13iT^{2} \) |
| 17 | \( 1 + 4.93iT - 17T^{2} \) |
| 19 | \( 1 + (4.77 - 4.77i)T - 19iT^{2} \) |
| 23 | \( 1 + 0.134T + 23T^{2} \) |
| 29 | \( 1 + (-2.17 + 2.17i)T - 29iT^{2} \) |
| 31 | \( 1 - 2.26T + 31T^{2} \) |
| 37 | \( 1 + (4.35 - 4.35i)T - 37iT^{2} \) |
| 41 | \( 1 - 3.34iT - 41T^{2} \) |
| 43 | \( 1 + (-2.70 + 2.70i)T - 43iT^{2} \) |
| 47 | \( 1 + 7.03iT - 47T^{2} \) |
| 53 | \( 1 + (-3.40 + 3.40i)T - 53iT^{2} \) |
| 59 | \( 1 + (0.107 + 0.107i)T + 59iT^{2} \) |
| 61 | \( 1 + (3.46 - 3.46i)T - 61iT^{2} \) |
| 67 | \( 1 + (-1.91 - 1.91i)T + 67iT^{2} \) |
| 71 | \( 1 + 9.32iT - 71T^{2} \) |
| 73 | \( 1 + 9.82T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + (8.80 + 8.80i)T + 83iT^{2} \) |
| 89 | \( 1 - 1.12iT - 89T^{2} \) |
| 97 | \( 1 - 6.10iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.52494598980273035734042665677, −13.27207769832610638529279602068, −12.26998048688253115146103074033, −11.73966445953958664428909777198, −10.07843319306108297265387148042, −8.728011313445029890795142825057, −7.07534610761965096033737705365, −6.24777356000852802619172177181, −4.28343193271277613432706933154, −3.32155890344846506280339010064,
2.98427119637115740645264346864, 4.19170700488718924150141187347, 6.04440659361145843904855354587, 7.07963053919030095553945104512, 8.529112511858611814176235981929, 10.42831081141775028994246912333, 11.12122641537857128972508783023, 12.54114490972908359872313545030, 13.10020845711868797343870742578, 14.23911900599099868506857146325