| L(s) = 1 | − 4.43·3-s + 6.56·5-s − 11.5i·7-s + 10.6·9-s + 5.05i·13-s − 29.1·15-s + 8.56i·17-s − 10.8i·19-s + 51.0i·21-s + 11.7·23-s + 18.0·25-s − 7.47·27-s − 20.2i·29-s − 36.4·31-s − 75.5i·35-s + ⋯ |
| L(s) = 1 | − 1.47·3-s + 1.31·5-s − 1.64i·7-s + 1.18·9-s + 0.389i·13-s − 1.94·15-s + 0.504i·17-s − 0.570i·19-s + 2.43i·21-s + 0.510·23-s + 0.721·25-s − 0.276·27-s − 0.697i·29-s − 1.17·31-s − 2.15i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.927i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.372 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9811191578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9811191578\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 4.43T + 9T^{2} \) |
| 5 | \( 1 - 6.56T + 25T^{2} \) |
| 7 | \( 1 + 11.5iT - 49T^{2} \) |
| 13 | \( 1 - 5.05iT - 169T^{2} \) |
| 17 | \( 1 - 8.56iT - 289T^{2} \) |
| 19 | \( 1 + 10.8iT - 361T^{2} \) |
| 23 | \( 1 - 11.7T + 529T^{2} \) |
| 29 | \( 1 + 20.2iT - 841T^{2} \) |
| 31 | \( 1 + 36.4T + 961T^{2} \) |
| 37 | \( 1 + 36.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 42.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 72.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 10.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 72.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 4.47T + 3.48e3T^{2} \) |
| 61 | \( 1 + 108. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 37.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 53.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 24.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 71.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 11.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 75.6T + 7.92e3T^{2} \) |
| 97 | \( 1 - 2.98T + 9.40e3T^{2} \) |
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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
−10.53986895209614651394800334115, −10.01321718344032432542646329171, −8.926191241860100055354530520760, −7.28430934516203665417317863677, −6.71428603210220761508290121020, −5.78459495889418125656315867711, −4.97846393406741564087569133275, −3.85733576990677011411810571631, −1.78819701777217851936192339916, −0.49134607663686539118241760768,
1.52212080844269529631277462917, 2.82070078764230200512506984891, 4.90559303342721170960610400305, 5.65000161639164385107107706129, 5.95996612902976166460769862948, 7.02308696374759842574199556503, 8.583684602554701720995334796139, 9.429163713417495942077310812540, 10.20819994161950083741146085016, 11.11809886385823390104844774280
