| L(s) = 1 | + 0.792i·3-s + 5-s − 4.70·7-s + 2.37·9-s + (−2.15 − 2.52i)11-s − 1.01i·13-s + 0.792i·15-s − 2.71i·17-s + 2.15·19-s − 3.72i·21-s + 5.04i·23-s + 25-s + 4.25i·27-s − 9.15i·29-s + 9.30i·31-s + ⋯ |
| L(s) = 1 | + 0.457i·3-s + 0.447·5-s − 1.77·7-s + 0.790·9-s + (−0.648 − 0.761i)11-s − 0.280i·13-s + 0.204i·15-s − 0.658i·17-s + 0.493·19-s − 0.813i·21-s + 1.05i·23-s + 0.200·25-s + 0.819i·27-s − 1.70i·29-s + 1.67i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.648 - 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.648 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8019428465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8019428465\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + (2.15 + 2.52i)T \) |
| good | 3 | \( 1 - 0.792iT - 3T^{2} \) |
| 7 | \( 1 + 4.70T + 7T^{2} \) |
| 13 | \( 1 + 1.01iT - 13T^{2} \) |
| 17 | \( 1 + 2.71iT - 17T^{2} \) |
| 19 | \( 1 - 2.15T + 19T^{2} \) |
| 23 | \( 1 - 5.04iT - 23T^{2} \) |
| 29 | \( 1 + 9.15iT - 29T^{2} \) |
| 31 | \( 1 - 9.30iT - 31T^{2} \) |
| 37 | \( 1 + 5.37T + 37T^{2} \) |
| 41 | \( 1 - 10.8iT - 41T^{2} \) |
| 43 | \( 1 - 2.55T + 43T^{2} \) |
| 47 | \( 1 - 1.87iT - 47T^{2} \) |
| 53 | \( 1 + 4.11T + 53T^{2} \) |
| 59 | \( 1 - 1.87iT - 59T^{2} \) |
| 61 | \( 1 - 7.13iT - 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 6.13iT - 71T^{2} \) |
| 73 | \( 1 - 11.8iT - 73T^{2} \) |
| 79 | \( 1 + 8.60T + 79T^{2} \) |
| 83 | \( 1 + 1.75T + 83T^{2} \) |
| 89 | \( 1 + 4.11T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
| 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
−9.138639800915764477227380966429, −8.111543297725928639364641159063, −7.22663556748446049403524518043, −6.61133897732998417019241206496, −5.80122125023239875205804334584, −5.19528858884355122144016081740, −4.10903315595950053919102889057, −3.22227422723914938157136636310, −2.75236162477076652597812349716, −1.15576168662604330240142940026,
0.25120615327085028879572847832, 1.71511233437883871699605145644, 2.57815887670650098889363525246, 3.53681144375040520369354672281, 4.38398068295496160834633732557, 5.44359957897612414910638731102, 6.20051342585772784761884476892, 6.94414629399119488367887640792, 7.22086130591040501970112152260, 8.297008407912849702885071451809
