| L(s) = 1 | + 6.78·7-s − 9.89i·11-s + 20.3·13-s − 19.1i·17-s + 12·19-s − 9.59i·23-s + 8.48i·29-s + 38·31-s − 6.78·37-s + 69.2i·41-s + 67.8·43-s − 76.7i·47-s − 3·49-s − 83.4i·59-s − 70·61-s + ⋯ |
| L(s) = 1 | + 0.968·7-s − 0.899i·11-s + 1.56·13-s − 1.12i·17-s + 0.631·19-s − 0.417i·23-s + 0.292i·29-s + 1.22·31-s − 0.183·37-s + 1.69i·41-s + 1.57·43-s − 1.63i·47-s − 0.0612·49-s − 1.41i·59-s − 1.14·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.842371181\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.842371181\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 6.78T + 49T^{2} \) |
| 11 | \( 1 + 9.89iT - 121T^{2} \) |
| 13 | \( 1 - 20.3T + 169T^{2} \) |
| 17 | \( 1 + 19.1iT - 289T^{2} \) |
| 19 | \( 1 - 12T + 361T^{2} \) |
| 23 | \( 1 + 9.59iT - 529T^{2} \) |
| 29 | \( 1 - 8.48iT - 841T^{2} \) |
| 31 | \( 1 - 38T + 961T^{2} \) |
| 37 | \( 1 + 6.78T + 1.36e3T^{2} \) |
| 41 | \( 1 - 69.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 67.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 76.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 + 83.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 70T + 3.72e3T^{2} \) |
| 67 | \( 1 + 108.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 118. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 13.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 30T + 6.24e3T^{2} \) |
| 83 | \( 1 - 134. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 32.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 94.9T + 9.40e3T^{2} \) |
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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
−8.213405699609342849224572174797, −7.75862007670208786766328506902, −6.68597902990991905775910321539, −6.05120399470625085635995113217, −5.20950698049703792464523122467, −4.52082221631845144632464170203, −3.51836289982596967452324126362, −2.75903552614264706230778618454, −1.47155557383070521239838784766, −0.70041212499084396180832474200,
1.12978006936586634143363347982, 1.76123179039136451161783926023, 2.94057182174775817946711839736, 4.06642463558453882537728383312, 4.48577917544116495308703653563, 5.64286915212832623973115177342, 6.08359473456281356657721735991, 7.15307537956096275937099993507, 7.79030615538427674075834517774, 8.462705275314710916105392772095
