| L(s) = 1 | − 1.55i·5-s − 12.3·7-s + 14.6i·11-s − 10.8·13-s − 28.9i·17-s − 3.60·19-s − 14.6i·23-s + 22.5·25-s + 28.1i·29-s − 8·31-s + 19.2i·35-s + 22.5·37-s + 25.1i·41-s + 53.1·43-s + 16.9i·47-s + ⋯ |
| L(s) = 1 | − 0.310i·5-s − 1.77·7-s + 1.33i·11-s − 0.831·13-s − 1.70i·17-s − 0.189·19-s − 0.638i·23-s + 0.903·25-s + 0.970i·29-s − 0.258·31-s + 0.549i·35-s + 0.609·37-s + 0.613i·41-s + 1.23·43-s + 0.361i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.160461694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.160461694\) |
| \(L(2)\) |
|
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 | \( 1 + 1.55iT - 25T^{2} \) |
| 7 | \( 1 + 12.3T + 49T^{2} \) |
| 11 | \( 1 - 14.6iT - 121T^{2} \) |
| 13 | \( 1 + 10.8T + 169T^{2} \) |
| 17 | \( 1 + 28.9iT - 289T^{2} \) |
| 19 | \( 1 + 3.60T + 361T^{2} \) |
| 23 | \( 1 + 14.6iT - 529T^{2} \) |
| 29 | \( 1 - 28.1iT - 841T^{2} \) |
| 31 | \( 1 + 8T + 961T^{2} \) |
| 37 | \( 1 - 22.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 25.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 53.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 16.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 84.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 91.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 13T + 3.72e3T^{2} \) |
| 67 | \( 1 - 41.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 16.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 71.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 46.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + 15.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 78.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 91.1T + 9.40e3T^{2} \) |
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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
−9.452213195057428505935304298205, −9.032267973163003126985144554510, −7.57873001370153118050136110382, −7.01452567492491256209256858840, −6.33491145572029935782786890845, −5.11226257044474976653757204288, −4.43796324458463718942592496890, −3.13399330374675070948933156806, −2.40265293466052000153383278375, −0.61633107363101671411790101312,
0.59289478156634947715199915868, 2.40488931889244623305361437806, 3.34493001838919030456233567788, 3.99676020802141707732509991083, 5.55308525064078227101288899382, 6.18507873942567468096268557501, 6.82329841934416079082101531541, 7.84352384581794493081423288282, 8.761927816063163961664142065428, 9.489460901027580149829772817263
