L(s) = 1 | + i·3-s + i·5-s + 2.30·7-s − 9-s + 0.945·11-s − 5.38·13-s − 15-s − 6.56i·17-s + 2.72·19-s + 2.30i·21-s + (0.880 − 4.71i)23-s − 25-s − i·27-s − 6.26·29-s + 5.84i·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.447i·5-s + 0.870·7-s − 0.333·9-s + 0.284·11-s − 1.49·13-s − 0.258·15-s − 1.59i·17-s + 0.626·19-s + 0.502i·21-s + (0.183 − 0.982i)23-s − 0.200·25-s − 0.192i·27-s − 1.16·29-s + 1.04i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.240034795\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.240034795\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (-0.880 + 4.71i)T \) |
good | 7 | \( 1 - 2.30T + 7T^{2} \) |
| 11 | \( 1 - 0.945T + 11T^{2} \) |
| 13 | \( 1 + 5.38T + 13T^{2} \) |
| 17 | \( 1 + 6.56iT - 17T^{2} \) |
| 19 | \( 1 - 2.72T + 19T^{2} \) |
| 29 | \( 1 + 6.26T + 29T^{2} \) |
| 31 | \( 1 - 5.84iT - 31T^{2} \) |
| 37 | \( 1 - 6.23iT - 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 + 8.79T + 43T^{2} \) |
| 47 | \( 1 - 9.81iT - 47T^{2} \) |
| 53 | \( 1 - 12.2iT - 53T^{2} \) |
| 59 | \( 1 - 5.84iT - 59T^{2} \) |
| 61 | \( 1 - 10.7iT - 61T^{2} \) |
| 67 | \( 1 + 6.05T + 67T^{2} \) |
| 71 | \( 1 - 9.51iT - 71T^{2} \) |
| 73 | \( 1 - 5.20T + 73T^{2} \) |
| 79 | \( 1 + 2.04T + 79T^{2} \) |
| 83 | \( 1 + 3.94T + 83T^{2} \) |
| 89 | \( 1 - 2.77iT - 89T^{2} \) |
| 97 | \( 1 - 9.24iT - 97T^{2} \) |
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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
−8.464070544829935937234632245176, −7.43309555716533039964503598686, −7.33978042633663393039485832234, −6.27566690559991299185033937639, −5.29781771006857089479042894223, −4.82065346186837151319497724057, −4.19899020700744073760891776004, −2.94293539665229835649042790983, −2.55820533938923316454901634034, −1.19142328926741830686266380352,
0.32317890641237859629158327884, 1.70233852669278532340557303652, 2.05621757983320660918469516800, 3.41784787261276304659144719599, 4.20180885189638258827436735634, 5.14859668489458673441837604943, 5.57362383184254235708023936277, 6.49028110320547015744752790070, 7.39325123084236762906127694551, 7.80759682440678629395658309288