| L(s) = 1 | + 2.87·5-s + 10.7i·7-s + 8i·11-s + 15.7·13-s + 15.8·17-s − 1.07i·19-s + 21.4i·23-s − 16.7·25-s + 40.0·29-s − 9.20i·31-s + 30.9i·35-s − 9.97·37-s + 51.5·41-s − 12.7i·43-s + 1.54i·47-s + ⋯ |
| L(s) = 1 | + 0.575·5-s + 1.53i·7-s + 0.727i·11-s + 1.20·13-s + 0.932·17-s − 0.0564i·19-s + 0.934i·23-s − 0.668·25-s + 1.38·29-s − 0.296i·31-s + 0.883i·35-s − 0.269·37-s + 1.25·41-s − 0.297i·43-s + 0.0328i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.520057070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.520057070\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.87T + 25T^{2} \) |
| 7 | \( 1 - 10.7iT - 49T^{2} \) |
| 11 | \( 1 - 8iT - 121T^{2} \) |
| 13 | \( 1 - 15.7T + 169T^{2} \) |
| 17 | \( 1 - 15.8T + 289T^{2} \) |
| 19 | \( 1 + 1.07iT - 361T^{2} \) |
| 23 | \( 1 - 21.4iT - 529T^{2} \) |
| 29 | \( 1 - 40.0T + 841T^{2} \) |
| 31 | \( 1 + 9.20iT - 961T^{2} \) |
| 37 | \( 1 + 9.97T + 1.36e3T^{2} \) |
| 41 | \( 1 - 51.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 12.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 1.54iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 28.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 11.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 1.54T + 3.72e3T^{2} \) |
| 67 | \( 1 - 43.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 84.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 105.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 73.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 12.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 33.1T + 7.92e3T^{2} \) |
| 97 | \( 1 + 69.1T + 9.40e3T^{2} \) |
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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
−9.054213183908672197258738379998, −8.364903186440208531186357927332, −7.57372082406465520325797026033, −6.48624259329399199057735513419, −5.78321000097589566936303548721, −5.35893720147484978949575568287, −4.18981380019324130692173851844, −3.09752884079536812195061242394, −2.21380112919572159025418023691, −1.28222422495197200853400674894,
0.67470697754731414144971149967, 1.40352237203214399235963239424, 2.88514247226324530234553887850, 3.76547639970374169194530130508, 4.47586428813686605267412972933, 5.64199636300016200786843013282, 6.28030351013936701754893634209, 7.02951039503943435086421218707, 7.949963821655565008772251230659, 8.512118417419494703575445235589
