| L(s) = 1 | + (−1 − i)3-s + (1 − 2i)5-s + (−1 + i)7-s − i·9-s − 6i·11-s + (−1 + i)13-s + (−3 + i)15-s + (1 + i)17-s + 4·19-s + 2·21-s + (5 + 5i)23-s + (−3 − 4i)25-s + (−4 + 4i)27-s + 8i·29-s + 2i·31-s + ⋯ |
| L(s) = 1 | + (−0.577 − 0.577i)3-s + (0.447 − 0.894i)5-s + (−0.377 + 0.377i)7-s − 0.333i·9-s − 1.80i·11-s + (−0.277 + 0.277i)13-s + (−0.774 + 0.258i)15-s + (0.242 + 0.242i)17-s + 0.917·19-s + 0.436·21-s + (1.04 + 1.04i)23-s + (−0.600 − 0.800i)25-s + (−0.769 + 0.769i)27-s + 1.48i·29-s + 0.359i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0898 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0898 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.701510 - 0.641101i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.701510 - 0.641101i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1 + 2i)T \) |
| good | 3 | \( 1 + (1 + i)T + 3iT^{2} \) |
| 7 | \( 1 + (1 - i)T - 7iT^{2} \) |
| 11 | \( 1 + 6iT - 11T^{2} \) |
| 13 | \( 1 + (1 - i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1 - i)T + 17iT^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (-5 - 5i)T + 23iT^{2} \) |
| 29 | \( 1 - 8iT - 29T^{2} \) |
| 31 | \( 1 - 2iT - 31T^{2} \) |
| 37 | \( 1 + (5 + 5i)T + 37iT^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + (-3 - 3i)T + 43iT^{2} \) |
| 47 | \( 1 + (-7 + 7i)T - 47iT^{2} \) |
| 53 | \( 1 + (1 - i)T - 53iT^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + (7 - 7i)T - 67iT^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 + (-9 + 9i)T - 73iT^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + (5 + 5i)T + 83iT^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (3 + 3i)T + 97iT^{2} \) |
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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
−12.60973931900205726092992753786, −11.84122163525417159032652226561, −10.86354213555925750386849642049, −9.332624203429572220075531583050, −8.764469932337465673096805489680, −7.26102270821270025689402728881, −5.95913764679941854902623536848, −5.37164686868675023044945954145, −3.35039622973022336506892140932, −1.07660708243586040684472500000,
2.53210305175926958911430710758, 4.31479746165326124901097683483, 5.45099286768068832275337042046, 6.81083599745482305637885064434, 7.62442994351652495618553157266, 9.608294248494044934125423362380, 10.07485291133370897386298538780, 10.92281188240218209976996052749, 12.02170692663250276549328285355, 13.12677797387634396171387640247
