| L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + 2i·7-s + i·8-s − 9-s − 5.23·11-s − i·12-s + 4.85i·13-s + 2·14-s + 16-s + 7.85i·17-s + i·18-s + 2.76·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.755i·7-s + 0.353i·8-s − 0.333·9-s − 1.57·11-s − 0.288i·12-s + 1.34i·13-s + 0.534·14-s + 0.250·16-s + 1.90i·17-s + 0.235i·18-s + 0.634·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3614281976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3614281976\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 - 4.85iT - 13T^{2} \) |
| 17 | \( 1 - 7.85iT - 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 - 3.70T + 31T^{2} \) |
| 37 | \( 1 + 2.14iT - 37T^{2} \) |
| 41 | \( 1 + 6.09T + 41T^{2} \) |
| 43 | \( 1 - 1.23iT - 43T^{2} \) |
| 47 | \( 1 - 4.76iT - 47T^{2} \) |
| 53 | \( 1 + 8.56iT - 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 + 8.85T + 61T^{2} \) |
| 67 | \( 1 + 9.70iT - 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 3.14iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 1.38T + 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
| 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
−8.860830221034468107862183161889, −8.420359719228031152383046520796, −7.67455531559021896038709865974, −6.43068509265896515109340524609, −5.80491862142006246606089942477, −4.87991577982101179458425478992, −4.35814978071044102626754047313, −3.32984324459851185681060694895, −2.52187506163899983044044301425, −1.70728067262587050533389121658,
0.11521953712764448075823316956, 1.08991769812352152234029935817, 2.74356135971675945942037262791, 3.23898963569154067218533385062, 4.63525716303569165153779198291, 5.30077673570284776819161951684, 5.78810828557661930102692055087, 6.94513805350540014349616274164, 7.54652755054198399085911945035, 7.74320071405945645296412778011
