| L(s) = 1 | − 3.86i·5-s + 3.73i·7-s + 1.03·11-s − 4.46·13-s − 1.79i·17-s + 1.73i·19-s − 8.76·23-s − 9.92·25-s − 7.72i·29-s − 7.46i·31-s + 14.4·35-s − 0.464·37-s + 7.72i·41-s + 0.535i·43-s + 4.62·47-s + ⋯ |
| L(s) = 1 | − 1.72i·5-s + 1.41i·7-s + 0.312·11-s − 1.23·13-s − 0.434i·17-s + 0.397i·19-s − 1.82·23-s − 1.98·25-s − 1.43i·29-s − 1.34i·31-s + 2.43·35-s − 0.0762·37-s + 1.20i·41-s + 0.0817i·43-s + 0.674·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3717954275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3717954275\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 3.86iT - 5T^{2} \) |
| 7 | \( 1 - 3.73iT - 7T^{2} \) |
| 11 | \( 1 - 1.03T + 11T^{2} \) |
| 13 | \( 1 + 4.46T + 13T^{2} \) |
| 17 | \( 1 + 1.79iT - 17T^{2} \) |
| 19 | \( 1 - 1.73iT - 19T^{2} \) |
| 23 | \( 1 + 8.76T + 23T^{2} \) |
| 29 | \( 1 + 7.72iT - 29T^{2} \) |
| 31 | \( 1 + 7.46iT - 31T^{2} \) |
| 37 | \( 1 + 0.464T + 37T^{2} \) |
| 41 | \( 1 - 7.72iT - 41T^{2} \) |
| 43 | \( 1 - 0.535iT - 43T^{2} \) |
| 47 | \( 1 - 4.62T + 47T^{2} \) |
| 53 | \( 1 - 3.58iT - 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 6.26iT - 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 3.92T + 73T^{2} \) |
| 79 | \( 1 - 4.80iT - 79T^{2} \) |
| 83 | \( 1 - 2.07T + 83T^{2} \) |
| 89 | \( 1 + 1.79iT - 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
| 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
−9.220207457546641892834296354527, −8.025896335713027600125690980472, −7.81348796564509402950401054947, −6.11969137705100033316852160407, −5.72340102619177249711081065459, −4.76452222679945403365421342063, −4.19242323428476113845165167264, −2.59449880349643795372450009826, −1.73006087471869684524593458834, −0.12956246680418703808014698963,
1.79994655036323348597449368478, 2.97748860867132492979997480728, 3.73220579314558851810244513061, 4.60362482453938919280932612821, 5.89286934689672001095346581120, 6.79429916450995450170649735382, 7.22777681258713777197601575574, 7.77442974069517781535884713316, 9.016830191581566202281026416719, 10.12110240642347628165865829331
