L(s) = 1 | + (−1 + 1.41i)3-s + (−1.00 − 2.82i)9-s + 6·11-s − 5.65i·17-s + 8.48i·19-s + 5·25-s + (5.00 + 1.41i)27-s + (−6 + 8.48i)33-s + 11.3i·41-s + 8.48i·43-s + 7·49-s + (8.00 + 5.65i)51-s + (−12 − 8.48i)57-s − 6·59-s − 8.48i·67-s + ⋯ |
L(s) = 1 | + (−0.577 + 0.816i)3-s + (−0.333 − 0.942i)9-s + 1.80·11-s − 1.37i·17-s + 1.94i·19-s + 25-s + (0.962 + 0.272i)27-s + (−1.04 + 1.47i)33-s + 1.76i·41-s + 1.29i·43-s + 49-s + (1.12 + 0.792i)51-s + (−1.58 − 1.12i)57-s − 0.781·59-s − 1.03i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.20190 + 0.622150i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20190 + 0.622150i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 - 1.41i)T \) |
good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 5.65iT - 17T^{2} \) |
| 19 | \( 1 - 8.48iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 11.3iT - 41T^{2} \) |
| 43 | \( 1 - 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 8.48iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 18T + 83T^{2} \) |
| 89 | \( 1 - 5.65iT - 89T^{2} \) |
| 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
−10.41028537391621499623050381498, −9.529679604819364368860346342313, −9.114235929380825439842956604773, −7.933233395447223951528066045261, −6.68738746488112880918032304464, −6.09965473708627903733411870985, −4.97038504677694939157559048920, −4.10097693297599374182001353842, −3.17748289583813502672208186173, −1.21325177578667372153513972808,
0.940193319874360338425704127048, 2.19796685980815850566575176915, 3.75030586513910626996503822543, 4.85296615641386915626758451398, 5.97665725329863433223916161740, 6.73788536351109334688842912040, 7.29678260320644648977236916626, 8.663013357009031343385744120378, 9.049635266734728437059057907380, 10.46033345850192554351588473598