L(s) = 1 | + (−1.36 − 1.36i)3-s + (0.866 − 0.5i)5-s + 2.73i·9-s + i·11-s + (−1.86 − 0.499i)15-s + (−0.366 − 0.366i)23-s + (0.499 − 0.866i)25-s + (2.36 − 2.36i)27-s − i·31-s + (1.36 − 1.36i)33-s + (−1.36 − 1.36i)37-s + (1.36 + 2.36i)45-s + (1 − i)47-s − i·49-s + (1 − i)53-s + ⋯ |
L(s) = 1 | + (−1.36 − 1.36i)3-s + (0.866 − 0.5i)5-s + 2.73i·9-s + i·11-s + (−1.86 − 0.499i)15-s + (−0.366 − 0.366i)23-s + (0.499 − 0.866i)25-s + (2.36 − 2.36i)27-s − i·31-s + (1.36 − 1.36i)33-s + (−1.36 − 1.36i)37-s + (1.36 + 2.36i)45-s + (1 − i)47-s − i·49-s + (1 − i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.473 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.473 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8238260165\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8238260165\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 11 | \( 1 - iT \) |
good | 3 | \( 1 + (1.36 + 1.36i)T + iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.366 + 0.366i)T + iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + iT - T^{2} \) |
| 37 | \( 1 + (1.36 + 1.36i)T + iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (-1 + i)T - iT^{2} \) |
| 53 | \( 1 + (-1 + i)T - iT^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 71 | \( 1 + 1.73iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - iT - T^{2} \) |
| 97 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
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
−8.354966574490130060218613218875, −7.51645910372355878745417187935, −6.90946353560199157889544809793, −6.32056243753819416547568577285, −5.47505734883305686562763179559, −5.15651593624672119896194799625, −4.12506895786280931140142993512, −2.21895088532058530696036994157, −1.91819321434550660751914295908, −0.64287471204707743279962651127,
1.20395041384012170671144681015, 2.90167105056054528659212652085, 3.62448002906489498602673664194, 4.56097601816196122610776822627, 5.37619916064942311692399323869, 5.84667250934985024488891245195, 6.44078641957222857156920400055, 7.21372299239001030401981284688, 8.635227140235580539271911087453, 9.125276276354412162330659714198