L(s) = 1 | + (0.331 + 1.37i)2-s − 2.13i·3-s + (−1.78 + 0.910i)4-s + (1.94 + 1.10i)5-s + (2.93 − 0.707i)6-s + (2.35 − 1.19i)7-s + (−1.84 − 2.14i)8-s − 1.56·9-s + (−0.874 + 3.03i)10-s + 2.33i·11-s + (1.94 + 3.80i)12-s + 1.09·13-s + (2.42 + 2.84i)14-s + (2.35 − 4.15i)15-s + (2.34 − 3.24i)16-s − 4.98·17-s + ⋯ |
L(s) = 1 | + (0.234 + 0.972i)2-s − 1.23i·3-s + (−0.890 + 0.455i)4-s + (0.869 + 0.493i)5-s + (1.19 − 0.288i)6-s + (0.891 − 0.453i)7-s + (−0.650 − 0.759i)8-s − 0.520·9-s + (−0.276 + 0.961i)10-s + 0.703i·11-s + (0.561 + 1.09i)12-s + 0.302·13-s + (0.649 + 0.760i)14-s + (0.608 − 1.07i)15-s + (0.585 − 0.810i)16-s − 1.20·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.28043 + 0.277346i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28043 + 0.277346i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.331 - 1.37i)T \) |
| 5 | \( 1 + (-1.94 - 1.10i)T \) |
| 7 | \( 1 + (-2.35 + 1.19i)T \) |
good | 3 | \( 1 + 2.13iT - 3T^{2} \) |
| 11 | \( 1 - 2.33iT - 11T^{2} \) |
| 13 | \( 1 - 1.09T + 13T^{2} \) |
| 17 | \( 1 + 4.98T + 17T^{2} \) |
| 19 | \( 1 + 2.57T + 19T^{2} \) |
| 23 | \( 1 + 6.04T + 23T^{2} \) |
| 29 | \( 1 - 0.561T + 29T^{2} \) |
| 31 | \( 1 + 6.59T + 31T^{2} \) |
| 37 | \( 1 + 5.49iT - 37T^{2} \) |
| 41 | \( 1 - 8.48iT - 41T^{2} \) |
| 43 | \( 1 - 1.32T + 43T^{2} \) |
| 47 | \( 1 - 9.74iT - 47T^{2} \) |
| 53 | \( 1 + 8.58iT - 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 - 0.620iT - 61T^{2} \) |
| 67 | \( 1 + 4.71T + 67T^{2} \) |
| 71 | \( 1 - 11.9iT - 71T^{2} \) |
| 73 | \( 1 + 9.96T + 73T^{2} \) |
| 79 | \( 1 + 10.6iT - 79T^{2} \) |
| 83 | \( 1 - 3.86iT - 83T^{2} \) |
| 89 | \( 1 + 2.82iT - 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−13.28742984125756306767836442527, −12.76239899315327204181053550522, −11.37128760097359032571605168977, −10.01721466094714763178737240981, −8.661540292170428244781915637149, −7.56876963662897004933914483559, −6.82377359839085400061651675108, −5.89017666218991051316235524621, −4.37483987094450817747511022240, −1.97856124390435057807884014007,
2.08474065639691169435157347492, 3.95082214716718244241299046618, 4.95493947249890316574534112561, 5.89379517535069694946361037134, 8.589120372318897719218431254504, 9.053340188376516624295720107823, 10.23364342602938280543219177879, 10.87764898525537671549700588250, 11.88514197098269191055204383442, 13.11858504653943726155799155187