| L(s) = 1 | + (−2 + i)5-s − 4i·7-s + 4·11-s − 4i·13-s − 6i·17-s + 4·19-s + 4i·23-s + (3 − 4i)25-s − 4·29-s + (4 + 8i)35-s + 4i·37-s − 8·41-s − 12i·47-s − 9·49-s + 2i·53-s + ⋯ |
| L(s) = 1 | + (−0.894 + 0.447i)5-s − 1.51i·7-s + 1.20·11-s − 1.10i·13-s − 1.45i·17-s + 0.917·19-s + 0.834i·23-s + (0.600 − 0.800i)25-s − 0.742·29-s + (0.676 + 1.35i)35-s + 0.657i·37-s − 1.24·41-s − 1.75i·47-s − 1.28·49-s + 0.274i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.964049 - 0.595815i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.964049 - 0.595815i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 - i)T \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 16iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−11.45981637780546412981609238709, −10.36872163673205174332565656535, −9.635242661206010463284466035687, −8.303461874670366343514258387297, −7.25463565131204283669373720379, −6.95770653309807617448312120485, −5.26728959389484267402735654426, −3.97981951892835348623994668646, −3.25471245534137786584865613970, −0.850804233762541821193714525302,
1.78704108518056960329219141797, 3.51848699303163267815961963257, 4.56077714954417616388435248870, 5.81361923301138332816845453477, 6.76676678404704570031497016309, 8.067672584663729312772467769997, 8.900102290329128880791534907300, 9.393688658630087900026978748972, 10.93914575936212787026276062008, 11.93288746169175770265723102064
