L(s) = 1 | + 0.618·2-s − 1.61·4-s + 1.61·5-s + 7-s − 2.23·8-s + 1.00·10-s − 4.23·13-s + 0.618·14-s + 1.85·16-s − 3.47·17-s − 6.47·19-s − 2.61·20-s − 5.70·23-s − 2.38·25-s − 2.61·26-s − 1.61·28-s − 0.236·29-s + 3·31-s + 5.61·32-s − 2.14·34-s + 1.61·35-s + 6·37-s − 4.00·38-s − 3.61·40-s − 9.85·41-s + 11.4·43-s − 3.52·46-s + ⋯ |
L(s) = 1 | + 0.437·2-s − 0.809·4-s + 0.723·5-s + 0.377·7-s − 0.790·8-s + 0.316·10-s − 1.17·13-s + 0.165·14-s + 0.463·16-s − 0.842·17-s − 1.48·19-s − 0.585·20-s − 1.19·23-s − 0.476·25-s − 0.513·26-s − 0.305·28-s − 0.0438·29-s + 0.538·31-s + 0.993·32-s − 0.368·34-s + 0.273·35-s + 0.986·37-s − 0.648·38-s − 0.572·40-s − 1.53·41-s + 1.74·43-s − 0.520·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.594163089\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.594163089\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 5 | \( 1 - 1.61T + 5T^{2} \) |
| 13 | \( 1 + 4.23T + 13T^{2} \) |
| 17 | \( 1 + 3.47T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 + 5.70T + 23T^{2} \) |
| 29 | \( 1 + 0.236T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 9.85T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 5.14T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 - 4.61T + 59T^{2} \) |
| 61 | \( 1 - 7.94T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 4.52T + 71T^{2} \) |
| 73 | \( 1 + 7.38T + 73T^{2} \) |
| 79 | \( 1 - 6.09T + 79T^{2} \) |
| 83 | \( 1 - 8.47T + 83T^{2} \) |
| 89 | \( 1 + 1.32T + 89T^{2} \) |
| 97 | \( 1 + 5.29T + 97T^{2} \) |
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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
−8.023296800587184906066091542012, −7.09041381057994825473531748576, −6.29147935917347475812068352835, −5.71705252697177004067140388189, −5.02244501162605671229909619516, −4.31028648014258481909082558584, −3.86018435185160707546338804520, −2.44778420218167146732059833435, −2.13129471143881045187875902753, −0.56785005415385041085852845725,
0.56785005415385041085852845725, 2.13129471143881045187875902753, 2.44778420218167146732059833435, 3.86018435185160707546338804520, 4.31028648014258481909082558584, 5.02244501162605671229909619516, 5.71705252697177004067140388189, 6.29147935917347475812068352835, 7.09041381057994825473531748576, 8.023296800587184906066091542012