L(s) = 1 | + (−2.27e5 + 1.29e5i)2-s + 1.77e8i·3-s + (3.49e10 − 5.91e10i)4-s + 5.55e12·5-s + (−2.30e13 − 4.03e13i)6-s + 2.49e15i·7-s + (−2.69e14 + 1.80e16i)8-s + 1.18e17·9-s + (−1.26e18 + 7.21e17i)10-s − 5.40e18i·11-s + (1.04e19 + 6.19e18i)12-s + 1.72e20·13-s + (−3.24e20 − 5.69e20i)14-s + 9.84e20i·15-s + (−2.27e21 − 4.13e21i)16-s + 8.60e21·17-s + ⋯ |
L(s) = 1 | + (−0.868 + 0.495i)2-s + 0.457i·3-s + (0.508 − 0.860i)4-s + 1.45·5-s + (−0.226 − 0.397i)6-s + 1.53i·7-s + (−0.0149 + 0.999i)8-s + 0.790·9-s + (−1.26 + 0.721i)10-s − 0.972i·11-s + (0.393 + 0.232i)12-s + 1.53·13-s + (−0.760 − 1.33i)14-s + 0.666i·15-s + (−0.482 − 0.875i)16-s + 0.611·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 - 0.860i)\, \overline{\Lambda}(37-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+18) \, L(s)\cr =\mathstrut & (0.508 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{37}{2})\) |
\(\approx\) |
\(2.121132357\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.121132357\) |
\(L(19)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.27e5 - 1.29e5i)T \) |
good | 3 | \( 1 - 1.77e8iT - 1.50e17T^{2} \) |
| 5 | \( 1 - 5.55e12T + 1.45e25T^{2} \) |
| 7 | \( 1 - 2.49e15iT - 2.65e30T^{2} \) |
| 11 | \( 1 + 5.40e18iT - 3.09e37T^{2} \) |
| 13 | \( 1 - 1.72e20T + 1.26e40T^{2} \) |
| 17 | \( 1 - 8.60e21T + 1.97e44T^{2} \) |
| 19 | \( 1 + 1.88e23iT - 1.08e46T^{2} \) |
| 23 | \( 1 + 9.41e23iT - 1.05e49T^{2} \) |
| 29 | \( 1 + 4.55e25T + 4.42e52T^{2} \) |
| 31 | \( 1 + 5.01e26iT - 4.88e53T^{2} \) |
| 37 | \( 1 - 1.31e28T + 2.85e56T^{2} \) |
| 41 | \( 1 - 2.91e27T + 1.14e58T^{2} \) |
| 43 | \( 1 - 6.20e28iT - 6.38e58T^{2} \) |
| 47 | \( 1 + 4.91e29iT - 1.56e60T^{2} \) |
| 53 | \( 1 - 4.35e29T + 1.18e62T^{2} \) |
| 59 | \( 1 - 5.43e31iT - 5.63e63T^{2} \) |
| 61 | \( 1 + 3.68e31T + 1.87e64T^{2} \) |
| 67 | \( 1 - 9.33e32iT - 5.47e65T^{2} \) |
| 71 | \( 1 + 1.44e33iT - 4.41e66T^{2} \) |
| 73 | \( 1 - 2.43e33T + 1.20e67T^{2} \) |
| 79 | \( 1 - 4.92e33iT - 2.06e68T^{2} \) |
| 83 | \( 1 - 3.24e34iT - 1.22e69T^{2} \) |
| 89 | \( 1 - 9.08e34T + 1.50e70T^{2} \) |
| 97 | \( 1 + 1.10e36T + 3.34e71T^{2} \) |
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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
−16.33010754035220656565919390605, −15.21256456951806077891984968293, −13.45239696329506058791948450945, −11.04580455606564223851134795464, −9.544696956549945957588854189111, −8.707713639950289705747409159145, −6.30373292173909526867533883680, −5.42740470668675607452650908615, −2.52851057964172917842280157606, −1.15150936161162853110955585664,
1.19712069706608386776815285995, 1.66740501937398034412756259438, 3.81677890688655224351583529294, 6.41850581088994271442283702099, 7.71360685147866186813200502721, 9.763275073376664222134909156723, 10.50978308704405337276228630339, 12.72347022342638791370862836790, 13.79978355424930679831326956490, 16.45948575211506164857068844214