| L(s) = 1 | + 7.38i·5-s + 6.79·7-s + 6.11i·11-s + 16.7·13-s + 25.1i·17-s + 17.5·19-s − 14.3i·23-s − 29.5·25-s − 18.7i·29-s + 46.7·31-s + 50.2i·35-s − 49.5·37-s − 39.8i·41-s + 44.1·43-s + 33.2i·47-s + ⋯ |
| L(s) = 1 | + 1.47i·5-s + 0.971·7-s + 0.556i·11-s + 1.29·13-s + 1.48i·17-s + 0.926·19-s − 0.622i·23-s − 1.18·25-s − 0.644i·29-s + 1.50·31-s + 1.43i·35-s − 1.34·37-s − 0.971i·41-s + 1.02·43-s + 0.707i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.421406701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.421406701\) |
| \(L(2)\) |
|
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 | \( 1 - 7.38iT - 25T^{2} \) |
| 7 | \( 1 - 6.79T + 49T^{2} \) |
| 11 | \( 1 - 6.11iT - 121T^{2} \) |
| 13 | \( 1 - 16.7T + 169T^{2} \) |
| 17 | \( 1 - 25.1iT - 289T^{2} \) |
| 19 | \( 1 - 17.5T + 361T^{2} \) |
| 23 | \( 1 + 14.3iT - 529T^{2} \) |
| 29 | \( 1 + 18.7iT - 841T^{2} \) |
| 31 | \( 1 - 46.7T + 961T^{2} \) |
| 37 | \( 1 + 49.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 39.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 44.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 33.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 10.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 16.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 21.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 86.9T + 4.48e3T^{2} \) |
| 71 | \( 1 - 30.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 48.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 111.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 98.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 75.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 140.T + 9.40e3T^{2} \) |
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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
−9.921842667361943020745920377518, −8.662712567297944603464555358330, −8.046198687041203613630877256954, −7.18900427670604544728842313843, −6.40001222693920541785424652188, −5.67272943051017410210861917051, −4.38722346325682192016648124099, −3.56262717072128984461327562946, −2.47058542395754465591983217148, −1.37946819783779371226259316870,
0.811093035360350248122745251737, 1.48371398020774650587650462382, 3.09945690429562958854242759675, 4.25889810426806561611821590420, 5.12458009901427453653597143435, 5.56715999634401214279325152409, 6.84150286553685718228501531611, 7.931290258200520325852161512389, 8.466034165749420372348067986660, 9.092713254535688286378198215763
