| L(s) = 1 | − 2.94·3-s − 3.19i·5-s − 3.28·7-s + 5.69·9-s − 1.70·11-s − 1.45i·13-s + 9.43i·15-s − 4.77i·17-s + 0.209i·19-s + 9.68·21-s − 8.73i·23-s − 5.23·25-s − 7.93·27-s − 2.65i·29-s − 9.47i·31-s + ⋯ |
| L(s) = 1 | − 1.70·3-s − 1.43i·5-s − 1.24·7-s + 1.89·9-s − 0.514·11-s − 0.403i·13-s + 2.43i·15-s − 1.15i·17-s + 0.0481i·19-s + 2.11·21-s − 1.82i·23-s − 1.04·25-s − 1.52·27-s − 0.492i·29-s − 1.70i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4006003658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4006003658\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 + (3.65 + 4.86i)T \) |
| good | 3 | \( 1 + 2.94T + 3T^{2} \) |
| 5 | \( 1 + 3.19iT - 5T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 13 | \( 1 + 1.45iT - 13T^{2} \) |
| 17 | \( 1 + 4.77iT - 17T^{2} \) |
| 19 | \( 1 - 0.209iT - 19T^{2} \) |
| 23 | \( 1 + 8.73iT - 23T^{2} \) |
| 29 | \( 1 + 2.65iT - 29T^{2} \) |
| 31 | \( 1 + 9.47iT - 31T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 + 8.67iT - 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 - 5.11T + 53T^{2} \) |
| 59 | \( 1 + 1.65iT - 59T^{2} \) |
| 61 | \( 1 - 1.90iT - 61T^{2} \) |
| 67 | \( 1 + 1.90T + 67T^{2} \) |
| 71 | \( 1 + 9.51T + 71T^{2} \) |
| 73 | \( 1 - 6.10T + 73T^{2} \) |
| 79 | \( 1 + 3.62iT - 79T^{2} \) |
| 83 | \( 1 - 3.20T + 83T^{2} \) |
| 89 | \( 1 - 10.1iT - 89T^{2} \) |
| 97 | \( 1 - 10.1iT - 97T^{2} \) |
| 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
−8.630810070813340016609062707183, −7.52169994018789252382475370487, −6.73968597167131112896556818335, −5.95660357799745599690602364144, −5.36505933206883166042801024616, −4.73930031040628284872231638408, −3.89773486973395360050930667992, −2.38985158934333964396894771459, −0.64812975525307623404504073711, −0.29892658938483430021925306343,
1.54788389231097456760361166196, 3.06931035329527436269430739813, 3.70403509523629286739494326804, 4.96535250654400860950142002040, 5.79520051775789354376132380400, 6.37914472272911532598976387231, 6.91039159411066601291222414191, 7.46214900874181876867831138816, 8.823137622545310411010867426856, 9.978689499971102870583967163677
