L(s) = 1 | + 1.08i·3-s − 2.61·5-s + (2.61 + 0.414i)7-s + 1.82·9-s − 2·11-s + 4.77·13-s − 2.82i·15-s − 3.06i·17-s + 4.14i·19-s + (−0.448 + 2.82i)21-s − 7.65i·23-s + 1.82·25-s + 5.22i·27-s − 3.65i·29-s + 3.06·31-s + ⋯ |
L(s) = 1 | + 0.624i·3-s − 1.16·5-s + (0.987 + 0.156i)7-s + 0.609·9-s − 0.603·11-s + 1.32·13-s − 0.730i·15-s − 0.742i·17-s + 0.950i·19-s + (−0.0978 + 0.617i)21-s − 1.59i·23-s + 0.365·25-s + 1.00i·27-s − 0.679i·29-s + 0.549·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 - 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.695273170\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.695273170\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.61 - 0.414i)T \) |
good | 3 | \( 1 - 1.08iT - 3T^{2} \) |
| 5 | \( 1 + 2.61T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 4.77T + 13T^{2} \) |
| 17 | \( 1 + 3.06iT - 17T^{2} \) |
| 19 | \( 1 - 4.14iT - 19T^{2} \) |
| 23 | \( 1 + 7.65iT - 23T^{2} \) |
| 29 | \( 1 + 3.65iT - 29T^{2} \) |
| 31 | \( 1 - 3.06T + 31T^{2} \) |
| 37 | \( 1 + 7.65iT - 37T^{2} \) |
| 41 | \( 1 - 9.55iT - 41T^{2} \) |
| 43 | \( 1 - 3.65T + 43T^{2} \) |
| 47 | \( 1 - 7.39T + 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 8.47iT - 59T^{2} \) |
| 61 | \( 1 + 2.61T + 61T^{2} \) |
| 67 | \( 1 - 15.6T + 67T^{2} \) |
| 71 | \( 1 - 8.82iT - 71T^{2} \) |
| 73 | \( 1 + 12.6iT - 73T^{2} \) |
| 79 | \( 1 - 12.8iT - 79T^{2} \) |
| 83 | \( 1 - 11.5iT - 83T^{2} \) |
| 89 | \( 1 + 2.16iT - 89T^{2} \) |
| 97 | \( 1 - 13.5iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.299449892076462838408435771715, −8.361412571575006565471820151617, −7.993634732316498334834240230024, −7.18253685497728078378980625885, −6.07647449832501431376948565751, −5.06823273788843524656931880954, −4.24999325589679962123229259269, −3.81381112699151364045764988501, −2.47931110319869755161832613200, −0.966996403155837504908417080410,
0.914152049920310541659483260335, 1.91081330550495548329053808456, 3.41393214051260499467603210847, 4.15519827637522119062432080037, 5.02989925505887212840640539611, 6.06805929068293738088799217355, 7.13091802666184862770968333543, 7.59985511329114009026141204770, 8.266367623687600647946020162588, 8.866406158233120497860287717106