| L(s) = 1 | − 2.73·3-s + 4.73i·7-s + 4.46·9-s + 3.46i·11-s − 3.46·13-s + 3.46i·17-s + 2i·19-s − 12.9i·21-s + 2.19i·23-s − 3.99·27-s + 2.53·31-s − 9.46i·33-s + 6·37-s + 9.46·39-s + 9.46·41-s + ⋯ |
| L(s) = 1 | − 1.57·3-s + 1.78i·7-s + 1.48·9-s + 1.04i·11-s − 0.960·13-s + 0.840i·17-s + 0.458i·19-s − 2.82i·21-s + 0.457i·23-s − 0.769·27-s + 0.455·31-s − 1.64i·33-s + 0.986·37-s + 1.51·39-s + 1.47·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.200i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 + 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4796040108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4796040108\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.73T + 3T^{2} \) |
| 7 | \( 1 - 4.73iT - 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 - 2.19iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 2.53T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 9.46T + 41T^{2} \) |
| 43 | \( 1 - 0.196T + 43T^{2} \) |
| 47 | \( 1 - 2.19iT - 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 + 0.928iT - 61T^{2} \) |
| 67 | \( 1 + 0.196T + 67T^{2} \) |
| 71 | \( 1 + 16.3T + 71T^{2} \) |
| 73 | \( 1 - 6.39iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 1.26T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 + 14.3iT - 97T^{2} \) |
| 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
−9.812479511166085427985595742802, −9.305246622533048421563764212053, −8.166495549749953062709072936541, −7.30104970079061546936765018134, −6.25617107940457083617485891599, −5.83847237913057077148854493682, −5.04358314280262983916860828749, −4.36724675804943809310919661332, −2.70340260420984196844832641315, −1.65847103499722340426603297638,
0.28313718598906060654198519889, 1.01889646950838046798308843526, 2.94596198990756254493241844612, 4.29131285372188037807306536014, 4.74034609792205732734381498466, 5.76348645714371591265766943086, 6.54022699345926043927241430258, 7.23507883510770865120330548893, 7.85123639041775866273355199790, 9.208310862471528502564042528261
