L(s) = 1 | + i·2-s − i·3-s − 4-s + (−1.40 + 1.74i)5-s + 6-s − i·8-s − 9-s + (−1.74 − 1.40i)10-s + 1.67·11-s + i·12-s − 4.48i·13-s + (1.74 + 1.40i)15-s + 16-s + 7.61i·17-s − i·18-s − 4.48·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.627 + 0.778i)5-s + 0.408·6-s − 0.353i·8-s − 0.333·9-s + (−0.550 − 0.443i)10-s + 0.505·11-s + 0.288i·12-s − 1.24i·13-s + (0.449 + 0.362i)15-s + 0.250·16-s + 1.84i·17-s − 0.235i·18-s − 1.02·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.627 + 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.09370405113\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09370405113\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 5 | \( 1 + (1.40 - 1.74i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 + 4.48iT - 13T^{2} \) |
| 17 | \( 1 - 7.61iT - 17T^{2} \) |
| 19 | \( 1 + 4.48T + 19T^{2} \) |
| 23 | \( 1 - 0.482iT - 23T^{2} \) |
| 29 | \( 1 + 1.19T + 29T^{2} \) |
| 31 | \( 1 + 3.48T + 31T^{2} \) |
| 37 | \( 1 + 0.482iT - 37T^{2} \) |
| 41 | \( 1 + 12.0T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + 11.4iT - 47T^{2} \) |
| 53 | \( 1 + 9.57iT - 53T^{2} \) |
| 59 | \( 1 + 5.77T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 + 3.35iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 4.51T + 79T^{2} \) |
| 83 | \( 1 + 1.87iT - 83T^{2} \) |
| 89 | \( 1 - 3.35T + 89T^{2} \) |
| 97 | \( 1 + 17.7iT - 97T^{2} \) |
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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
−8.828279716399947086515166621972, −8.206104928331983933733303453040, −7.66544284883050530224874742462, −6.67547059792442862563309383545, −6.25714844075055341751395778341, −5.27396047126451044135273519603, −3.97744239849485532273953921276, −3.30753054084456600091580964765, −1.83133442830073518017343496669, −0.03747268631843508602171277101,
1.51650479241496825950742900242, 2.85719719758762009061822757496, 3.98330725366455158673658874402, 4.52666682955703933651577200749, 5.27412922615973924200389300581, 6.55235394770482996849472472922, 7.50904308391195649823151646530, 8.537402045350424494634657173465, 9.230691050566736377529007421764, 9.511846236764548719222211697192