L(s) = 1 | + 0.286i·3-s − 0.286i·7-s + 2.91·9-s + 4.26·11-s + 3.20i·13-s − 0.286i·17-s + 19-s + 0.0820·21-s + 0.936i·23-s + 1.69i·27-s − 2.26·29-s − 4.18·31-s + 1.22i·33-s − 8.67i·37-s − 0.917·39-s + ⋯ |
L(s) = 1 | + 0.165i·3-s − 0.108i·7-s + 0.972·9-s + 1.28·11-s + 0.888i·13-s − 0.0694i·17-s + 0.229·19-s + 0.0179·21-s + 0.195i·23-s + 0.326i·27-s − 0.421·29-s − 0.751·31-s + 0.212i·33-s − 1.42i·37-s − 0.146·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.053653533\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.053653533\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 0.286iT - 3T^{2} \) |
| 7 | \( 1 + 0.286iT - 7T^{2} \) |
| 11 | \( 1 - 4.26T + 11T^{2} \) |
| 13 | \( 1 - 3.20iT - 13T^{2} \) |
| 17 | \( 1 + 0.286iT - 17T^{2} \) |
| 23 | \( 1 - 0.936iT - 23T^{2} \) |
| 29 | \( 1 + 2.26T + 29T^{2} \) |
| 31 | \( 1 + 4.18T + 31T^{2} \) |
| 37 | \( 1 + 8.67iT - 37T^{2} \) |
| 41 | \( 1 - 1.08T + 41T^{2} \) |
| 43 | \( 1 - 4.42iT - 43T^{2} \) |
| 47 | \( 1 - 0.759iT - 47T^{2} \) |
| 53 | \( 1 - 4.42iT - 53T^{2} \) |
| 59 | \( 1 - 4.70T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 9.82iT - 67T^{2} \) |
| 71 | \( 1 - 4.83T + 71T^{2} \) |
| 73 | \( 1 - 10.4iT - 73T^{2} \) |
| 79 | \( 1 + 5.10T + 79T^{2} \) |
| 83 | \( 1 - 15.7iT - 83T^{2} \) |
| 89 | \( 1 - 9.75T + 89T^{2} \) |
| 97 | \( 1 - 9.61iT - 97T^{2} \) |
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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
−9.374274621990568686553775057224, −8.684254841581479412463612592128, −7.49436806210320418128641937317, −6.99044297766891802743986379254, −6.19780950302329501767272831110, −5.16710890266188848408535406145, −4.10565635326503537390454307865, −3.75214478955797858849550791251, −2.17164383396088408427077589074, −1.15942764922985240014791125707,
0.945721141435967371517106898310, 2.01635699316324936724975159383, 3.37006498760838785035399721635, 4.12133811345971645441563824884, 5.10969867883336478248094966466, 6.04356803875479508894617692000, 6.86471737345315234014819316682, 7.47599283759351528889419058937, 8.417690652998357140592606852738, 9.147468625404384210512509117878