| L(s) = 1 | − 2.02i·3-s + (−3.26 − 3.26i)7-s − 1.11·9-s + (−3.55 − 3.55i)11-s − 1.41·13-s + (−1.73 − 1.73i)17-s + (4.19 + 4.19i)19-s + (−6.63 + 6.63i)21-s + (0.177 − 0.177i)23-s − 3.82i·27-s + (1.63 − 1.63i)29-s + 8.15i·31-s + (−7.20 + 7.20i)33-s + 1.34·37-s + 2.87i·39-s + ⋯ |
| L(s) = 1 | − 1.17i·3-s + (−1.23 − 1.23i)7-s − 0.371·9-s + (−1.07 − 1.07i)11-s − 0.393·13-s + (−0.420 − 0.420i)17-s + (0.962 + 0.962i)19-s + (−1.44 + 1.44i)21-s + (0.0370 − 0.0370i)23-s − 0.735i·27-s + (0.304 − 0.304i)29-s + 1.46i·31-s + (−1.25 + 1.25i)33-s + 0.220·37-s + 0.460i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4883922729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4883922729\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.02iT - 3T^{2} \) |
| 7 | \( 1 + (3.26 + 3.26i)T + 7iT^{2} \) |
| 11 | \( 1 + (3.55 + 3.55i)T + 11iT^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 + (1.73 + 1.73i)T + 17iT^{2} \) |
| 19 | \( 1 + (-4.19 - 4.19i)T + 19iT^{2} \) |
| 23 | \( 1 + (-0.177 + 0.177i)T - 23iT^{2} \) |
| 29 | \( 1 + (-1.63 + 1.63i)T - 29iT^{2} \) |
| 31 | \( 1 - 8.15iT - 31T^{2} \) |
| 37 | \( 1 - 1.34T + 37T^{2} \) |
| 41 | \( 1 - 1.16iT - 41T^{2} \) |
| 43 | \( 1 - 1.04T + 43T^{2} \) |
| 47 | \( 1 + (4.25 - 4.25i)T - 47iT^{2} \) |
| 53 | \( 1 - 8.19iT - 53T^{2} \) |
| 59 | \( 1 + (-3.96 + 3.96i)T - 59iT^{2} \) |
| 61 | \( 1 + (7.22 + 7.22i)T + 61iT^{2} \) |
| 67 | \( 1 - 9.19T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + (5.86 + 5.86i)T + 73iT^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 12.0iT - 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 + (5.71 + 5.71i)T + 97iT^{2} \) |
| 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
−8.747444772799472026792019611903, −7.71172667259075759555178562051, −7.41230901301003851802403798606, −6.52857653862078806159385232202, −5.92011609255240573986947676015, −4.73023916697552593576603138263, −3.45560930030888439853779742956, −2.78458045793166578730935208032, −1.25163303681640666353147308302, −0.19497001278234848151030255961,
2.32985281556218319679069957615, 3.01790462360574497517617141211, 4.12891212570201744267113954025, 5.04350599587208568460847166635, 5.60405557423066396077499444956, 6.69361811918528027348843363199, 7.51129305869912923201853532774, 8.666407128165872200445115106587, 9.370922005054477564143777777373, 9.849788311359356470337384644783
