| L(s) = 1 | + (0.707 − 1.22i)2-s + (−1.5 + 0.866i)3-s + (−0.999 − 1.73i)4-s + 2.44i·6-s + (−3.97 + 5.76i)7-s − 2.82·8-s + (1.5 − 2.59i)9-s + (−1.64 − 2.85i)11-s + (2.99 + 1.73i)12-s − 7.72i·13-s + (4.24 + 8.94i)14-s + (−2.00 + 3.46i)16-s + (−10.9 + 6.30i)17-s + (−2.12 − 3.67i)18-s + (1.54 + 0.890i)19-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.5 + 0.288i)3-s + (−0.249 − 0.433i)4-s + 0.408i·6-s + (−0.567 + 0.823i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−0.149 − 0.259i)11-s + (0.249 + 0.144i)12-s − 0.594i·13-s + (0.303 + 0.638i)14-s + (−0.125 + 0.216i)16-s + (−0.642 + 0.371i)17-s + (−0.117 − 0.204i)18-s + (0.0812 + 0.0468i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.515538392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.515538392\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.707 + 1.22i)T \) |
| 3 | \( 1 + (1.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (3.97 - 5.76i)T \) |
| good | 11 | \( 1 + (1.64 + 2.85i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 7.72iT - 169T^{2} \) |
| 17 | \( 1 + (10.9 - 6.30i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-1.54 - 0.890i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-3.37 + 5.85i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 39.2T + 841T^{2} \) |
| 31 | \( 1 + (-9.46 + 5.46i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (17.1 - 29.7i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 18.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 77.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-11.1 - 6.44i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-25.8 - 44.7i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-97.7 + 56.4i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (22.7 + 13.1i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-9.95 - 17.2i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 87.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-52.7 + 30.4i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-17.7 + 30.8i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 46.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (47.4 + 27.4i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 45.7iT - 9.40e3T^{2} \) |
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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
−9.814098274830373722087866955835, −8.960079938669140677252344541959, −8.205395481324137311055481743242, −6.79533916147062242076225615180, −6.02066390415674446366576353858, −5.29029498948690758790231830890, −4.33514961480483320766548093281, −3.23742037180951737946421089918, −2.33041053714926577434407029990, −0.67881349049474286780492336722,
0.77786244306994645498414238925, 2.49657470704118397418356840207, 3.85609298142525543882781915298, 4.63132932817438856109461072305, 5.60509788752010013621022340296, 6.66899732687566109769206599690, 6.98265168425235259764336952423, 7.905662446674255794077279573290, 8.935543574968527952314654747894, 9.799311246000306454785628265718
