L(s) = 1 | + (−0.256 + 2.22i)5-s + 3.50·7-s + 1.92·11-s − 5.50i·13-s + 4.44·17-s − 7.00i·19-s + 1.10i·23-s + (−4.86 − 1.14i)25-s − 5.47i·29-s + 8.28i·31-s + (−0.900 + 7.78i)35-s − 0.778i·37-s − 2.44i·41-s + 9.55·43-s + 11.7i·47-s + ⋯ |
L(s) = 1 | + (−0.114 + 0.993i)5-s + 1.32·7-s + 0.581·11-s − 1.52i·13-s + 1.07·17-s − 1.60i·19-s + 0.229i·23-s + (−0.973 − 0.228i)25-s − 1.01i·29-s + 1.48i·31-s + (−0.152 + 1.31i)35-s − 0.127i·37-s − 0.381i·41-s + 1.45·43-s + 1.70i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0547i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.052617733\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.052617733\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.256 - 2.22i)T \) |
good | 7 | \( 1 - 3.50T + 7T^{2} \) |
| 11 | \( 1 - 1.92T + 11T^{2} \) |
| 13 | \( 1 + 5.50iT - 13T^{2} \) |
| 17 | \( 1 - 4.44T + 17T^{2} \) |
| 19 | \( 1 + 7.00iT - 19T^{2} \) |
| 23 | \( 1 - 1.10iT - 23T^{2} \) |
| 29 | \( 1 + 5.47iT - 29T^{2} \) |
| 31 | \( 1 - 8.28iT - 31T^{2} \) |
| 37 | \( 1 + 0.778iT - 37T^{2} \) |
| 41 | \( 1 + 2.44iT - 41T^{2} \) |
| 43 | \( 1 - 9.55T + 43T^{2} \) |
| 47 | \( 1 - 11.7iT - 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 9.78T + 59T^{2} \) |
| 61 | \( 1 + 3.45T + 61T^{2} \) |
| 67 | \( 1 + 5.45T + 67T^{2} \) |
| 71 | \( 1 - 4.25T + 71T^{2} \) |
| 73 | \( 1 + 7.27iT - 73T^{2} \) |
| 79 | \( 1 - 2.82iT - 79T^{2} \) |
| 83 | \( 1 - 4.25iT - 83T^{2} \) |
| 89 | \( 1 - 0.386iT - 89T^{2} \) |
| 97 | \( 1 + 9.29iT - 97T^{2} \) |
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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
−9.577292204081108477337372057335, −8.646034723140887516271510327895, −7.70588525255044730084307072921, −7.39934749863478987297508026339, −6.22805602060828749808691137394, −5.40068183590715197473533088834, −4.50333051494677495010837151911, −3.34752654960888040922538328254, −2.49898467695342993638960433727, −1.04387983910098404773703748460,
1.24989195828657484596674129926, 1.94971130945076969315863364004, 3.82171685788763515256557945315, 4.38326498501211271021601536375, 5.31341869306392045709954139306, 6.07336260438682478044592854739, 7.32015499040260345607658066836, 7.988804094208287739976449698882, 8.708774077165203001274183601176, 9.365366334841796305084902621999