| L(s) = 1 | + i·3-s − 1.11·5-s + (2.00 + 1.73i)7-s − 9-s − 4.26i·11-s + 2.13i·13-s − 1.11i·15-s − 7.30·17-s + 1.25·19-s + (−1.73 + 2.00i)21-s + (2.60 + 4.02i)23-s − 3.75·25-s − i·27-s − 4.69·29-s + 10.0i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.499·5-s + (0.756 + 0.654i)7-s − 0.333·9-s − 1.28i·11-s + 0.591i·13-s − 0.288i·15-s − 1.77·17-s + 0.287·19-s + (−0.377 + 0.436i)21-s + (0.542 + 0.840i)23-s − 0.750·25-s − 0.192i·27-s − 0.871·29-s + 1.81i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6269468950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6269468950\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 + (-2.00 - 1.73i)T \) |
| 23 | \( 1 + (-2.60 - 4.02i)T \) |
| good | 5 | \( 1 + 1.11T + 5T^{2} \) |
| 11 | \( 1 + 4.26iT - 11T^{2} \) |
| 13 | \( 1 - 2.13iT - 13T^{2} \) |
| 17 | \( 1 + 7.30T + 17T^{2} \) |
| 19 | \( 1 - 1.25T + 19T^{2} \) |
| 29 | \( 1 + 4.69T + 29T^{2} \) |
| 31 | \( 1 - 10.0iT - 31T^{2} \) |
| 37 | \( 1 - 3.75iT - 37T^{2} \) |
| 41 | \( 1 + 4.12iT - 41T^{2} \) |
| 43 | \( 1 + 4.35iT - 43T^{2} \) |
| 47 | \( 1 - 8.59iT - 47T^{2} \) |
| 53 | \( 1 + 6.83iT - 53T^{2} \) |
| 59 | \( 1 - 3.01iT - 59T^{2} \) |
| 61 | \( 1 + 5.54T + 61T^{2} \) |
| 67 | \( 1 + 7.79iT - 67T^{2} \) |
| 71 | \( 1 + 16.3T + 71T^{2} \) |
| 73 | \( 1 - 1.71iT - 73T^{2} \) |
| 79 | \( 1 + 6.31iT - 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.196885404692887749678047065625, −8.923833124903176261603637560715, −8.207532402334884452864458195980, −7.29182090012509401902199744687, −6.30629893072953766458603766334, −5.44980630462283396597241990507, −4.70126282849761918512466142257, −3.81943426288469055687336883601, −2.90273669427976035340339726879, −1.65231096734820260133674678923,
0.21781428700575603657234658922, 1.69922735222865516783854550555, 2.58695839186460630620599671532, 4.10470156100566407894408662446, 4.51168206390900029523998395444, 5.61849553718178591728865598826, 6.68326100228383520480603928663, 7.37859935797192888401909805344, 7.82766587193504137450205417318, 8.683218873497113474702454704438
