| L(s) = 1 | − 3·3-s + 6.63i·5-s + 8·7-s + 9·9-s + 3.31i·11-s + 4·13-s − 19.8i·15-s − 13.2i·17-s + 6·19-s − 24·21-s + 6.63i·23-s − 19·25-s − 27·27-s + 39.7i·29-s + 26·31-s + ⋯ |
| L(s) = 1 | − 3-s + 1.32i·5-s + 1.14·7-s + 9-s + 0.301i·11-s + 0.307·13-s − 1.32i·15-s − 0.780i·17-s + 0.315·19-s − 1.14·21-s + 0.288i·23-s − 0.760·25-s − 27-s + 1.37i·29-s + 0.838·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.367014701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.367014701\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 11 | \( 1 - 3.31iT \) |
| good | 5 | \( 1 - 6.63iT - 25T^{2} \) |
| 7 | \( 1 - 8T + 49T^{2} \) |
| 13 | \( 1 - 4T + 169T^{2} \) |
| 17 | \( 1 + 13.2iT - 289T^{2} \) |
| 19 | \( 1 - 6T + 361T^{2} \) |
| 23 | \( 1 - 6.63iT - 529T^{2} \) |
| 29 | \( 1 - 39.7iT - 841T^{2} \) |
| 31 | \( 1 - 26T + 961T^{2} \) |
| 37 | \( 1 - 30T + 1.36e3T^{2} \) |
| 41 | \( 1 + 13.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 42T + 1.84e3T^{2} \) |
| 47 | \( 1 - 86.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 59.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 66.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 12T + 3.72e3T^{2} \) |
| 67 | \( 1 + 2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 59.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 74T + 5.32e3T^{2} \) |
| 79 | \( 1 - 40T + 6.24e3T^{2} \) |
| 83 | \( 1 + 39.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 119. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 62T + 9.40e3T^{2} \) |
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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
−10.92620692475161324516011631602, −10.33943251028426868990002767479, −9.286202169935506125068647791610, −7.85515545791436087353982109763, −7.16560744760734825664589802660, −6.33835001495353085865767011517, −5.27938396910351851496202005640, −4.36622202926411974006768008516, −2.90725838550722139861647276043, −1.38267461020345039610994042751,
0.69800252997721015209033229666, 1.75239054810510256747324559388, 4.05142803184477132938254993222, 4.85540258468475393895855049036, 5.55876087665585514855833515599, 6.56829085631142286162290215295, 7.990771306290242187444338538061, 8.429260650455487585054064908035, 9.628932645442603423512893619920, 10.51810821558973494417326180412
