L(s) = 1 | + (1.40 + 0.118i)2-s + (1.97 + 0.335i)4-s + 3.46i·5-s + 7-s + (2.73 + 0.707i)8-s + (−0.411 + 4.88i)10-s − 3.31i·11-s + 3.10i·13-s + (1.40 + 0.118i)14-s + (3.77 + 1.32i)16-s − 1.40·17-s + 4.80i·19-s + (−1.16 + 6.82i)20-s + (0.394 − 4.67i)22-s + 8.79·23-s + ⋯ |
L(s) = 1 | + (0.996 + 0.0841i)2-s + (0.985 + 0.167i)4-s + 1.54i·5-s + 0.377·7-s + (0.968 + 0.249i)8-s + (−0.130 + 1.54i)10-s − 1.00i·11-s + 0.862i·13-s + (0.376 + 0.0317i)14-s + (0.943 + 0.330i)16-s − 0.340·17-s + 1.10i·19-s + (−0.259 + 1.52i)20-s + (0.0841 − 0.997i)22-s + 1.83·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.448280268\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.448280268\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.40 - 0.118i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 3.46iT - 5T^{2} \) |
| 11 | \( 1 + 3.31iT - 11T^{2} \) |
| 13 | \( 1 - 3.10iT - 13T^{2} \) |
| 17 | \( 1 + 1.40T + 17T^{2} \) |
| 19 | \( 1 - 4.80iT - 19T^{2} \) |
| 23 | \( 1 - 8.79T + 23T^{2} \) |
| 29 | \( 1 + 9.87iT - 29T^{2} \) |
| 31 | \( 1 + 7.83T + 31T^{2} \) |
| 37 | \( 1 - 5.42iT - 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 7.03iT - 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 6.51iT - 53T^{2} \) |
| 59 | \( 1 + 3.89iT - 59T^{2} \) |
| 61 | \( 1 + 9.42iT - 61T^{2} \) |
| 67 | \( 1 + 0.909iT - 67T^{2} \) |
| 71 | \( 1 + 6.42T + 71T^{2} \) |
| 73 | \( 1 - 1.56T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 0.370iT - 83T^{2} \) |
| 89 | \( 1 - 4.85T + 89T^{2} \) |
| 97 | \( 1 - 1.63T + 97T^{2} \) |
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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
−9.910959780625221594585750855058, −8.699118923988865007030270443278, −7.71892034159813349584680191664, −7.02916564581856971309701845577, −6.33197356328618178381304746721, −5.68048167930790194842162194235, −4.51938150377795619198732921532, −3.55090371428091367708515492776, −2.88704889539099894936266425209, −1.79612924101506294165866867126,
1.03338253282637387859277275967, 2.11280734128788328152834305664, 3.40263916232106973842280445810, 4.51140148021007400514079575525, 5.11934296508039697280992399236, 5.48866851559313780801247317751, 7.03966590338183921878234855276, 7.36025445287344087466080230790, 8.752024258288839323306186227905, 9.018013975864045637594091733781