| L(s) = 1 | − 4.83i·5-s + 9.71·7-s + 15.3i·11-s − 8.20·13-s + 10.4i·17-s − 14.3·19-s − 24.5i·23-s + 1.59·25-s − 31.1i·29-s − 60.5·31-s − 46.9i·35-s + 0.734·37-s − 46.9i·41-s + 74.4·43-s + 12.7i·47-s + ⋯ |
| L(s) = 1 | − 0.967i·5-s + 1.38·7-s + 1.39i·11-s − 0.631·13-s + 0.612i·17-s − 0.755·19-s − 1.06i·23-s + 0.0638·25-s − 1.07i·29-s − 1.95·31-s − 1.34i·35-s + 0.0198·37-s − 1.14i·41-s + 1.73·43-s + 0.270i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.913234818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.913234818\) |
| \(L(2)\) |
|
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 | \( 1 + 4.83iT - 25T^{2} \) |
| 7 | \( 1 - 9.71T + 49T^{2} \) |
| 11 | \( 1 - 15.3iT - 121T^{2} \) |
| 13 | \( 1 + 8.20T + 169T^{2} \) |
| 17 | \( 1 - 10.4iT - 289T^{2} \) |
| 19 | \( 1 + 14.3T + 361T^{2} \) |
| 23 | \( 1 + 24.5iT - 529T^{2} \) |
| 29 | \( 1 + 31.1iT - 841T^{2} \) |
| 31 | \( 1 + 60.5T + 961T^{2} \) |
| 37 | \( 1 - 0.734T + 1.36e3T^{2} \) |
| 41 | \( 1 + 46.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 74.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 12.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 36.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 89.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 40.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 80.4T + 4.48e3T^{2} \) |
| 71 | \( 1 + 0.687iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 27.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 126.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 13.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 168. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 64.0T + 9.40e3T^{2} \) |
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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
−8.407980479445746246763041624640, −7.70607260524419861165589526775, −7.09893830014232235702085189808, −5.97710271699336450670742901259, −5.04065820664116766388727933071, −4.61305722470596661256906255627, −3.95885241234707793449368171772, −2.18464530232793666730435923012, −1.81777262140854154221678575219, −0.45477475133111401019947685224,
1.08261589095569899755964916113, 2.21716747250709965234818206573, 3.09111011647874817706832994613, 3.96228065543618519408728789897, 5.04436462638059782136662342730, 5.61699025324083338673485421590, 6.55597277688693327535385495997, 7.43633433975748484072383197130, 7.82776310288107422998393201709, 8.814522802194607674001000160642
