L(s) = 1 | + (−1.51 + 0.848i)3-s − 0.936i·5-s − 2.20i·7-s + (1.56 − 2.56i)9-s − 1.69·11-s − 4.27·13-s + (0.794 + 1.41i)15-s − 1.12i·17-s + 4.34i·19-s + (1.87 + 3.33i)21-s − 7.24·23-s + 4.12·25-s + (−0.185 + 5.19i)27-s − 5.73i·29-s + 5.03i·31-s + ⋯ |
L(s) = 1 | + (−0.871 + 0.489i)3-s − 0.418i·5-s − 0.834i·7-s + (0.520 − 0.853i)9-s − 0.511·11-s − 1.18·13-s + (0.205 + 0.365i)15-s − 0.272i·17-s + 0.996i·19-s + (0.408 + 0.727i)21-s − 1.51·23-s + 0.824·25-s + (−0.0357 + 0.999i)27-s − 1.06i·29-s + 0.904i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 - 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.489 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4254404204\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4254404204\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.51 - 0.848i)T \) |
good | 5 | \( 1 + 0.936iT - 5T^{2} \) |
| 7 | \( 1 + 2.20iT - 7T^{2} \) |
| 11 | \( 1 + 1.69T + 11T^{2} \) |
| 13 | \( 1 + 4.27T + 13T^{2} \) |
| 17 | \( 1 + 1.12iT - 17T^{2} \) |
| 19 | \( 1 - 4.34iT - 19T^{2} \) |
| 23 | \( 1 + 7.24T + 23T^{2} \) |
| 29 | \( 1 + 5.73iT - 29T^{2} \) |
| 31 | \( 1 - 5.03iT - 31T^{2} \) |
| 37 | \( 1 - 9.06T + 37T^{2} \) |
| 41 | \( 1 - 4iT - 41T^{2} \) |
| 43 | \( 1 - 7.73iT - 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 - 14.2iT - 53T^{2} \) |
| 59 | \( 1 + 6.41T + 59T^{2} \) |
| 61 | \( 1 + 8.01T + 61T^{2} \) |
| 67 | \( 1 - 10.3iT - 67T^{2} \) |
| 71 | \( 1 - 1.58T + 71T^{2} \) |
| 73 | \( 1 + 6.24T + 73T^{2} \) |
| 79 | \( 1 + 10.6iT - 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 - 13.1iT - 89T^{2} \) |
| 97 | \( 1 + 7.12T + 97T^{2} \) |
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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
−9.962213951306912451706146603932, −9.155500655783291132061961488736, −7.914445124919659461357578550475, −7.40751041381674242668203879624, −6.30340385834296757591130375575, −5.61114487922240319309818552197, −4.56444225643101064231917587999, −4.19629553468232227909978788771, −2.78766707785536200602814344686, −1.15763697674847594662764635182,
0.20381880622588018624747023870, 2.03210645065166888492299407935, 2.76275522447189131124108264576, 4.33621664078350212561044918890, 5.23445880174164049925563759612, 5.86084554919766453596256449572, 6.78113170152171699877785889561, 7.43425488762647555128145590140, 8.246063815228748172231396339809, 9.270827634772880716882884451355