| L(s) = 1 | + (−2 − i)5-s + (−5 + 5i)13-s + (5 + 5i)17-s + (3 + 4i)25-s + 4i·29-s + (5 + 5i)37-s − 8·41-s + 7i·49-s + (−5 + 5i)53-s − 12·61-s + (15 − 5i)65-s + (5 − 5i)73-s + (−5 − 15i)85-s − 16i·89-s + (5 + 5i)97-s + ⋯ |
| L(s) = 1 | + (−0.894 − 0.447i)5-s + (−1.38 + 1.38i)13-s + (1.21 + 1.21i)17-s + (0.600 + 0.800i)25-s + 0.742i·29-s + (0.821 + 0.821i)37-s − 1.24·41-s + i·49-s + (−0.686 + 0.686i)53-s − 1.53·61-s + (1.86 − 0.620i)65-s + (0.585 − 0.585i)73-s + (−0.542 − 1.62i)85-s − 1.69i·89-s + (0.507 + 0.507i)97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0898 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0898 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.558495 + 0.611121i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.558495 + 0.611121i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (5 - 5i)T - 13iT^{2} \) |
| 17 | \( 1 + (-5 - 5i)T + 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23iT^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + (-5 - 5i)T + 37iT^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (5 - 5i)T - 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-5 + 5i)T - 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 + 16iT - 89T^{2} \) |
| 97 | \( 1 + (-5 - 5i)T + 97iT^{2} \) |
| 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
−10.64065210830210825438694943800, −9.715288814539549451072271140271, −8.921850869445639689069105511047, −7.968249838097597296014444644572, −7.31861043615295832735158602451, −6.28591592061112047681363820206, −5.01419410160804988396575696349, −4.28295712761428366738200999132, −3.17301110087739539798135539875, −1.56151718046754628350246714915,
0.43842981696271729148507804072, 2.63212306855336792251199475316, 3.44409810353912618084281220687, 4.75168108136968396186086502134, 5.56676128846362329260603883788, 6.90350327093729244211903579518, 7.65505867975109223610531894634, 8.147513325142824767473202160957, 9.541155496830400545204402493216, 10.13501475744345699642418941512
