L(s) = 1 | + (−1.45 − 2.51i)5-s + (−2.64 − 0.0146i)7-s + (−1.08 − 0.628i)11-s + 5.32i·13-s + (−0.880 + 1.52i)17-s + (−6.69 + 3.86i)19-s + (−4.43 + 2.55i)23-s + (−1.72 + 2.98i)25-s − 8.74i·29-s + (2.18 + 1.26i)31-s + (3.80 + 6.68i)35-s + (−3.66 − 6.34i)37-s + 3.09·41-s + 1.15·43-s + (−3.44 − 5.96i)47-s + ⋯ |
L(s) = 1 | + (−0.649 − 1.12i)5-s + (−0.999 − 0.00554i)7-s + (−0.328 − 0.189i)11-s + 1.47i·13-s + (−0.213 + 0.369i)17-s + (−1.53 + 0.887i)19-s + (−0.924 + 0.533i)23-s + (−0.344 + 0.597i)25-s − 1.62i·29-s + (0.392 + 0.226i)31-s + (0.643 + 1.12i)35-s + (−0.602 − 1.04i)37-s + 0.483·41-s + 0.176·43-s + (−0.502 − 0.869i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00512268 + 0.0327290i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00512268 + 0.0327290i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.64 + 0.0146i)T \) |
good | 5 | \( 1 + (1.45 + 2.51i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.08 + 0.628i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5.32iT - 13T^{2} \) |
| 17 | \( 1 + (0.880 - 1.52i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (6.69 - 3.86i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.43 - 2.55i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 8.74iT - 29T^{2} \) |
| 31 | \( 1 + (-2.18 - 1.26i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.66 + 6.34i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 3.09T + 41T^{2} \) |
| 43 | \( 1 - 1.15T + 43T^{2} \) |
| 47 | \( 1 + (3.44 + 5.96i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (11.3 + 6.52i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.13 - 5.43i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.19 + 1.26i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.689 - 1.19i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8.65iT - 71T^{2} \) |
| 73 | \( 1 + (-4.38 - 2.52i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.63 + 13.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 3.75T + 83T^{2} \) |
| 89 | \( 1 + (-3.55 - 6.15i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 4.03iT - 97T^{2} \) |
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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
−10.32959842935929874409989796450, −9.455382708344982085739244302978, −8.638196713960344353598759553857, −7.918202839389931413073621032287, −6.64858452200421761251986828326, −5.83834910218109487143624243634, −4.39494227622066586700036367625, −3.84291344854037314404424052900, −2.01788949915396124314362925492, −0.01843287789945307748189353886,
2.68047352724672640901787581209, 3.35472777584512901439894320538, 4.68636039556424261327184303217, 6.11985085098221497831333275374, 6.81624637599728445724339130534, 7.70642797278550737118657923232, 8.650904961220627925739622163867, 9.858618249681369488257432619647, 10.62346401820799689161214493460, 11.10037311830133396065048352037