L(s) = 1 | + 1.41i·2-s − 2.00·4-s − 2.23i·5-s − 2.82i·8-s + 3.16·10-s − 4.47i·11-s + 6.32·13-s + 4.00·16-s + 2.82i·17-s + 4.47i·20-s + 6.32·22-s − 5.65i·23-s − 5.00·25-s + 8.94i·26-s − 4.47i·29-s + ⋯ |
L(s) = 1 | + 0.999i·2-s − 1.00·4-s − 0.999i·5-s − 1.00i·8-s + 1.00·10-s − 1.34i·11-s + 1.75·13-s + 1.00·16-s + 0.685i·17-s + 1.00i·20-s + 1.34·22-s − 1.17i·23-s − 1.00·25-s + 1.75i·26-s − 0.830i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.20927\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20927\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 4.47iT - 11T^{2} \) |
| 13 | \( 1 - 6.32T + 13T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 5.65iT - 23T^{2} \) |
| 29 | \( 1 + 4.47iT - 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 6.32T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 - 11.3iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 4.47iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−11.45205635204913498411136669714, −10.41567656321262078899303283324, −9.168766467204766204175063320286, −8.454277839923177052897084485410, −8.024631287508971036416228629671, −6.28824334314141833793795985233, −5.91383316039994105522590822993, −4.59283801964257390722453744422, −3.59317917963346676892765135471, −0.950694344106414543605407735459,
1.71300875044962520823166294165, 3.09845057699046422838914145274, 4.06656873665894839718196062428, 5.40656403214343627806968857069, 6.69464312490420043592651254763, 7.76241569416037228531766315761, 8.931330139761770118320368626966, 9.863965520311408513233796740054, 10.57253708896704443997451271860, 11.42901318668806618943603842066