L(s) = 1 | − i·2-s + (0.5 + 1.65i)3-s − 4-s + (−1.91 − 1.15i)5-s + (1.65 − 0.5i)6-s − 3.21i·7-s + i·8-s + (−2.5 + 1.65i)9-s + (−1.15 + 1.91i)10-s + 4.31i·11-s + (−0.5 − 1.65i)12-s − 3.31·13-s − 3.21·14-s + (0.964 − 3.75i)15-s + 16-s − 3.21·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.288 + 0.957i)3-s − 0.5·4-s + (−0.855 − 0.518i)5-s + (0.677 − 0.204i)6-s − 1.21i·7-s + 0.353i·8-s + (−0.833 + 0.552i)9-s + (−0.366 + 0.604i)10-s + 1.30i·11-s + (−0.144 − 0.478i)12-s − 0.919·13-s − 0.860·14-s + (0.249 − 0.968i)15-s + 0.250·16-s − 0.780·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00263167 + 0.0132869i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00263167 + 0.0132869i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-0.5 - 1.65i)T \) |
| 5 | \( 1 + (1.91 + 1.15i)T \) |
| 19 | \( 1 + (4.31 + 0.605i)T \) |
good | 7 | \( 1 + 3.21iT - 7T^{2} \) |
| 11 | \( 1 - 4.31iT - 11T^{2} \) |
| 13 | \( 1 + 3.31T + 13T^{2} \) |
| 17 | \( 1 + 3.21T + 17T^{2} \) |
| 23 | \( 1 + 7.04T + 23T^{2} \) |
| 29 | \( 1 - 8.25T + 29T^{2} \) |
| 31 | \( 1 + 2.61iT - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 + 3.82iT - 43T^{2} \) |
| 47 | \( 1 + 8.86T + 47T^{2} \) |
| 53 | \( 1 + iT - 53T^{2} \) |
| 59 | \( 1 + 5.64T + 59T^{2} \) |
| 61 | \( 1 + 2.31T + 61T^{2} \) |
| 67 | \( 1 - 7T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 1.81iT - 73T^{2} \) |
| 79 | \( 1 - 15.3iT - 79T^{2} \) |
| 83 | \( 1 - 6.43T + 83T^{2} \) |
| 89 | \( 1 - 2.61T + 89T^{2} \) |
| 97 | \( 1 - 3.36T + 97T^{2} \) |
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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
−10.13700261174321690766830371510, −9.744564707978898317759210574555, −8.553196871378370688288082891266, −7.86264184899074168520465513250, −6.77702140641985545285378373670, −4.84125416569397465383804606521, −4.46052939774307538651339336658, −3.68641635899840647004933818209, −2.18169764214780526648115895234, −0.00694900430621809062069547377,
2.34738329700134853225669742141, 3.40399614788660971277999040548, 4.89468271924918610855581267881, 6.25153727190767914828905057625, 6.52383308061750493639521824016, 7.85042226926372679102918584335, 8.344124116165754773610701463557, 8.974194019868231812811137978973, 10.34826826214954746560511649173, 11.56252279602673804222330572058