| L(s) = 1 | − 2·2-s + 2·4-s − 5-s − 3·7-s + 2·10-s + 6·13-s + 6·14-s − 4·16-s + 3·17-s + 19-s − 2·20-s − 4·23-s − 4·25-s − 12·26-s − 6·28-s − 10·29-s + 2·31-s + 8·32-s − 6·34-s + 3·35-s + 8·37-s − 2·38-s − 8·41-s + 43-s + 8·46-s − 3·47-s + 2·49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.447·5-s − 1.13·7-s + 0.632·10-s + 1.66·13-s + 1.60·14-s − 16-s + 0.727·17-s + 0.229·19-s − 0.447·20-s − 0.834·23-s − 4/5·25-s − 2.35·26-s − 1.13·28-s − 1.85·29-s + 0.359·31-s + 1.41·32-s − 1.02·34-s + 0.507·35-s + 1.31·37-s − 0.324·38-s − 1.24·41-s + 0.152·43-s + 1.17·46-s − 0.437·47-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20691 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20691 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5104910745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5104910745\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.75498740859363, −15.46098417190397, −14.70768979511995, −13.80383729129521, −13.43534822766410, −12.95416072827254, −12.19051481256727, −11.45549372514651, −11.23798175959389, −10.39561622728319, −10.01758332968994, −9.396885538583134, −9.073475582496539, −8.189502554416206, −7.988731184558783, −7.297078986296944, −6.607528270516365, −6.052250699158225, −5.480741427285755, −4.264436383379720, −3.717964749602516, −3.147213188159605, −2.057550628367009, −1.292698686384929, −0.4006813149579238,
0.4006813149579238, 1.292698686384929, 2.057550628367009, 3.147213188159605, 3.717964749602516, 4.264436383379720, 5.480741427285755, 6.052250699158225, 6.607528270516365, 7.297078986296944, 7.988731184558783, 8.189502554416206, 9.073475582496539, 9.396885538583134, 10.01758332968994, 10.39561622728319, 11.23798175959389, 11.45549372514651, 12.19051481256727, 12.95416072827254, 13.43534822766410, 13.80383729129521, 14.70768979511995, 15.46098417190397, 15.75498740859363
