L(s) = 1 | + (1.72 − 0.114i)3-s + 2.40·5-s − 3.48i·7-s + (2.97 − 0.396i)9-s − 1.00·11-s − 1.58i·13-s + (4.15 − 0.276i)15-s − 3.02i·17-s − 3.52·19-s + (−0.399 − 6.01i)21-s + 5.89i·23-s + 0.782·25-s + (5.09 − 1.02i)27-s − 6.29i·29-s + 10.0i·31-s + ⋯ |
L(s) = 1 | + (0.997 − 0.0662i)3-s + 1.07·5-s − 1.31i·7-s + (0.991 − 0.132i)9-s − 0.303·11-s − 0.440i·13-s + (1.07 − 0.0712i)15-s − 0.734i·17-s − 0.809·19-s + (−0.0871 − 1.31i)21-s + 1.22i·23-s + 0.156·25-s + (0.980 − 0.197i)27-s − 1.16i·29-s + 1.80i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.39520 - 0.862323i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.39520 - 0.862323i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.72 + 0.114i)T \) |
| 67 | \( 1 + (-5.94 - 5.62i)T \) |
good | 5 | \( 1 - 2.40T + 5T^{2} \) |
| 7 | \( 1 + 3.48iT - 7T^{2} \) |
| 11 | \( 1 + 1.00T + 11T^{2} \) |
| 13 | \( 1 + 1.58iT - 13T^{2} \) |
| 17 | \( 1 + 3.02iT - 17T^{2} \) |
| 19 | \( 1 + 3.52T + 19T^{2} \) |
| 23 | \( 1 - 5.89iT - 23T^{2} \) |
| 29 | \( 1 + 6.29iT - 29T^{2} \) |
| 31 | \( 1 - 10.0iT - 31T^{2} \) |
| 37 | \( 1 + 6.52T + 37T^{2} \) |
| 41 | \( 1 - 4.16T + 41T^{2} \) |
| 43 | \( 1 - 5.62iT - 43T^{2} \) |
| 47 | \( 1 - 2.23iT - 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 2.62iT - 59T^{2} \) |
| 61 | \( 1 - 12.1iT - 61T^{2} \) |
| 71 | \( 1 - 9.00iT - 71T^{2} \) |
| 73 | \( 1 + 0.635T + 73T^{2} \) |
| 79 | \( 1 + 9.44iT - 79T^{2} \) |
| 83 | \( 1 + 7.80iT - 83T^{2} \) |
| 89 | \( 1 + 15.8iT - 89T^{2} \) |
| 97 | \( 1 - 8.49iT - 97T^{2} \) |
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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
−10.18713123618936924745022124421, −9.392902771106922349405676816681, −8.521477512458227069506939280516, −7.51823705343184607794173756265, −6.96066218694083662959442106371, −5.78702372908487965111209482676, −4.62690076699224570690274397432, −3.59474174771469984200069281468, −2.49655848217785737193262325200, −1.27700966497499060363781127523,
2.01608585232388351683009257272, 2.40115315340786020459770343925, 3.81430644351178491481346939976, 5.04948263896568375212179185887, 5.99339110244929193166701985794, 6.79935187105351517699213095238, 8.116614509083773716533802419642, 8.757770004293140423631172769354, 9.350820016449875475829063859443, 10.15375896012206083909900463513