| L(s) = 1 | − 3.19i·3-s − 5-s + (−2.59 − 0.525i)7-s − 7.23·9-s + 3.34·11-s − 3.90·13-s + 3.19i·15-s + 2.92i·17-s − 6.33i·19-s + (−1.68 + 8.29i)21-s + 3.44i·23-s + 25-s + 13.5i·27-s + 2.68i·29-s − 2.52·31-s + ⋯ |
| L(s) = 1 | − 1.84i·3-s − 0.447·5-s + (−0.980 − 0.198i)7-s − 2.41·9-s + 1.00·11-s − 1.08·13-s + 0.826i·15-s + 0.710i·17-s − 1.45i·19-s + (−0.366 + 1.81i)21-s + 0.718i·23-s + 0.200·25-s + 2.61i·27-s + 0.497i·29-s − 0.453·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00384 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00384 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1247196831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1247196831\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (2.59 + 0.525i)T \) |
| good | 3 | \( 1 + 3.19iT - 3T^{2} \) |
| 11 | \( 1 - 3.34T + 11T^{2} \) |
| 13 | \( 1 + 3.90T + 13T^{2} \) |
| 17 | \( 1 - 2.92iT - 17T^{2} \) |
| 19 | \( 1 + 6.33iT - 19T^{2} \) |
| 23 | \( 1 - 3.44iT - 23T^{2} \) |
| 29 | \( 1 - 2.68iT - 29T^{2} \) |
| 31 | \( 1 + 2.52T + 31T^{2} \) |
| 37 | \( 1 - 4.70iT - 37T^{2} \) |
| 41 | \( 1 - 5.59iT - 41T^{2} \) |
| 43 | \( 1 + 8.62T + 43T^{2} \) |
| 47 | \( 1 + 0.506T + 47T^{2} \) |
| 53 | \( 1 + 11.4iT - 53T^{2} \) |
| 59 | \( 1 - 0.802iT - 59T^{2} \) |
| 61 | \( 1 - 7.97T + 61T^{2} \) |
| 67 | \( 1 + 6.37T + 67T^{2} \) |
| 71 | \( 1 - 1.11iT - 71T^{2} \) |
| 73 | \( 1 - 5.91iT - 73T^{2} \) |
| 79 | \( 1 + 10.5iT - 79T^{2} \) |
| 83 | \( 1 + 4.29iT - 83T^{2} \) |
| 89 | \( 1 - 2.00iT - 89T^{2} \) |
| 97 | \( 1 + 6.06iT - 97T^{2} \) |
| 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
−9.031331292385742801145525539948, −8.246123334487560119443245732850, −7.27982373656473381111318529025, −6.84873205184262557181666565874, −6.25253548437802871639931261911, −5.05502461359746236130743250960, −3.57260608096207376358440518525, −2.62809313592916042453190702165, −1.38287507660435935777690649677, −0.05526172139875803334748769371,
2.64374014412449823369604284381, 3.64279293710427448889571562758, 4.18019931897558932358322412771, 5.18933042967183081955866115353, 6.02517995420538041950772888190, 7.07811941673218601211096956385, 8.275829004376892705586709867139, 9.159277594962225574049176077901, 9.631513514037702809795436500958, 10.23719921332571393492921628598
