L(s) = 1 | + 2-s + (−1.61 − 0.618i)3-s + 4-s + (−1.61 − 0.618i)6-s + (2.61 + 0.381i)7-s + 8-s + (2.23 + 2.00i)9-s + 4.47i·11-s + (−1.61 − 0.618i)12-s − 3.23·13-s + (2.61 + 0.381i)14-s + 16-s + 0.763i·17-s + (2.23 + 2.00i)18-s − 0.472i·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.934 − 0.356i)3-s + 0.5·4-s + (−0.660 − 0.252i)6-s + (0.989 + 0.144i)7-s + 0.353·8-s + (0.745 + 0.666i)9-s + 1.34i·11-s + (−0.467 − 0.178i)12-s − 0.897·13-s + (0.699 + 0.102i)14-s + 0.250·16-s + 0.185i·17-s + (0.527 + 0.471i)18-s − 0.108i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 - 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.965733381\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.965733381\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (1.61 + 0.618i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.61 - 0.381i)T \) |
good | 11 | \( 1 - 4.47iT - 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 - 0.763iT - 17T^{2} \) |
| 19 | \( 1 + 0.472iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 5.70iT - 29T^{2} \) |
| 31 | \( 1 - 7.23iT - 31T^{2} \) |
| 37 | \( 1 - 5.23iT - 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 + 12.9iT - 43T^{2} \) |
| 47 | \( 1 - 2.47iT - 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 + 2.76iT - 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 2.76iT - 71T^{2} \) |
| 73 | \( 1 - 6.76T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 - 16.6iT - 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 - 5.23T + 97T^{2} \) |
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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
−10.38915026511409324821017334251, −9.262296412539246462404144837987, −8.032953152702273250427033985563, −7.14341076757609441107859371447, −6.76708913248062965026807378408, −5.27831764470113514120003743645, −5.07615583130982832862373006489, −4.13897608242807353134322195402, −2.45181275391618129208314372198, −1.45036470134578584843213020945,
0.869774493084748360691975202743, 2.52919806393723827691262263543, 3.89689964513329360040216535979, 4.65329896354591816174690246371, 5.51207586273090543294886826879, 6.09766930998894013556334516414, 7.21616657403126395516736003839, 7.963950558645050391548514109716, 9.120511303013982189695048769186, 10.07708060492348293895185593878