L(s) = 1 | + 2.61i·3-s + 2.61i·5-s + 7-s − 3.82·9-s + 2.16i·11-s − 0.448i·13-s − 6.82·15-s − 7.65·17-s + 4.77i·19-s + 2.61i·21-s + 6.82·23-s − 1.82·25-s − 2.16i·27-s − 9.55i·29-s + 5.65·31-s + ⋯ |
L(s) = 1 | + 1.50i·3-s + 1.16i·5-s + 0.377·7-s − 1.27·9-s + 0.652i·11-s − 0.124i·13-s − 1.76·15-s − 1.85·17-s + 1.09i·19-s + 0.570i·21-s + 1.42·23-s − 0.365·25-s − 0.416i·27-s − 1.77i·29-s + 1.01·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(-1.33907i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-1.33907i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 - 2.61iT - 3T^{2} \) |
| 5 | \( 1 - 2.61iT - 5T^{2} \) |
| 11 | \( 1 - 2.16iT - 11T^{2} \) |
| 13 | \( 1 + 0.448iT - 13T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 19 | \( 1 - 4.77iT - 19T^{2} \) |
| 23 | \( 1 - 6.82T + 23T^{2} \) |
| 29 | \( 1 + 9.55iT - 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 - 5.22iT - 37T^{2} \) |
| 41 | \( 1 + 3.65T + 41T^{2} \) |
| 43 | \( 1 + 2.16iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 10.4iT - 53T^{2} \) |
| 59 | \( 1 + 0.448iT - 59T^{2} \) |
| 61 | \( 1 - 12.1iT - 61T^{2} \) |
| 67 | \( 1 - 3.06iT - 67T^{2} \) |
| 71 | \( 1 - 2.34T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 2.34T + 79T^{2} \) |
| 83 | \( 1 + 13.0iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−10.25575179449264081903307301670, −10.06881798996065468389637484727, −9.026753572817571750567891152910, −8.177748116240146983730449928112, −7.00836544098240935592014370262, −6.25135149903560284433546345555, −4.97410486289329159355585511665, −4.33101888893388265923544817982, −3.34162436164300955239520925699, −2.29153649005241271250802659531,
0.64624383288167329175721718935, 1.65434548895765472842795280294, 2.85505933486014992029168659074, 4.57121584337150750018229324129, 5.25256743639434861428355054113, 6.55201049442649774859874188828, 6.98217485171851867756057303295, 8.120470726735648571684275540269, 8.743911282028836808520674395252, 9.221371339709143518115580769649