L(s) = 1 | + (−0.195 + 0.980i)2-s + (−1.38 − 1.38i)3-s + (−0.923 − 0.382i)4-s + (0.707 − 0.707i)5-s + (1.63 − 1.08i)6-s + (0.555 − 0.831i)8-s + 2.84i·9-s + (0.555 + 0.831i)10-s + (0.541 − 0.541i)11-s + (0.750 + 1.81i)12-s + (1.17 + 1.17i)13-s − 1.96·15-s + (0.707 + 0.707i)16-s + (−2.79 − 0.555i)18-s + (0.707 + 0.707i)19-s + (−0.923 + 0.382i)20-s + ⋯ |
L(s) = 1 | + (−0.195 + 0.980i)2-s + (−1.38 − 1.38i)3-s + (−0.923 − 0.382i)4-s + (0.707 − 0.707i)5-s + (1.63 − 1.08i)6-s + (0.555 − 0.831i)8-s + 2.84i·9-s + (0.555 + 0.831i)10-s + (0.541 − 0.541i)11-s + (0.750 + 1.81i)12-s + (1.17 + 1.17i)13-s − 1.96·15-s + (0.707 + 0.707i)16-s + (−2.79 − 0.555i)18-s + (0.707 + 0.707i)19-s + (−0.923 + 0.382i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7119192569\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7119192569\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.195 - 0.980i)T \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
| 19 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + (1.38 + 1.38i)T + iT^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 13 | \( 1 + (-1.17 - 1.17i)T + iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-0.275 + 0.275i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-1.38 + 1.38i)T - iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 67 | \( 1 + (-0.785 - 0.785i)T + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + 1.66T + T^{2} \) |
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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
−9.417482131684073477569483840676, −8.539841001918001526974162247694, −7.927888283968342764145413753267, −6.84166778660510457779676963622, −6.41558980335162284389234556991, −5.74443686125037583899267073696, −5.17581705596652995301242303538, −4.08449859349570615917540018194, −1.75446954098929112750711308888, −1.01818786768436410565491404853,
1.13544775185211412368775257157, 2.96398884726267826035963654230, 3.70109148455252281112921292852, 4.63321075548402588870009607755, 5.48925415890466596535621308166, 6.08577747489134654234015033965, 7.12885404528320080250841384646, 8.579856455121930263349840914986, 9.495210091199874261481811935203, 9.812609859896217708980456774838