L(s) = 1 | + (−1.41 + i)3-s − 2.23·5-s + (−1.41 + 2.23i)7-s + (1.00 − 2.82i)9-s − 2.82i·11-s + 2.82·13-s + (3.16 − 2.23i)15-s − 4i·17-s − 6.32i·19-s + (−0.236 − 4.57i)21-s − 6.32·23-s + 5.00·25-s + (1.41 + 5.00i)27-s − 5.65i·29-s + 6.32i·31-s + ⋯ |
L(s) = 1 | + (−0.816 + 0.577i)3-s − 0.999·5-s + (−0.534 + 0.845i)7-s + (0.333 − 0.942i)9-s − 0.852i·11-s + 0.784·13-s + (0.816 − 0.577i)15-s − 0.970i·17-s − 1.45i·19-s + (−0.0515 − 0.998i)21-s − 1.31·23-s + 1.00·25-s + (0.272 + 0.962i)27-s − 1.05i·29-s + 1.13i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0515 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0515 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.326399 - 0.309997i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.326399 - 0.309997i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.41 - i)T \) |
| 5 | \( 1 + 2.23T \) |
| 7 | \( 1 + (1.41 - 2.23i)T \) |
good | 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 6.32iT - 19T^{2} \) |
| 23 | \( 1 + 6.32T + 23T^{2} \) |
| 29 | \( 1 + 5.65iT - 29T^{2} \) |
| 31 | \( 1 - 6.32iT - 31T^{2} \) |
| 37 | \( 1 + 8.94iT - 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 - 4.47iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 6.32T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 4.47iT - 67T^{2} \) |
| 71 | \( 1 - 14.1iT - 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 10iT - 83T^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 + 8.48T + 97T^{2} \) |
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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
−11.31923640379870030920776440977, −10.17767870251408460216230377640, −9.120930142551513912617257782025, −8.450496328999637824240724383088, −7.07002161307769202260037883691, −6.14497997128675635152451107494, −5.20327006137785901705366343402, −4.04647717804194184249663204598, −3.01178223375518413082995415499, −0.33568394706146756358022019481,
1.50232950144928434982096801616, 3.64215200831933250644247612596, 4.45504106164841867106587527947, 5.92585070541606187749454263499, 6.74333857413770863880778594693, 7.68083142370235439428184218203, 8.303636568340436300827376613108, 9.987294973844655775929823288980, 10.54532195437980666965315515310, 11.49967331399361969426154407308