| L(s) = 1 | − 1.08i·3-s − 1.08i·5-s + 7-s + 1.82·9-s + 5.22i·11-s + 6.30i·13-s − 1.17·15-s + 3.65·17-s + 4.14i·19-s − 1.08i·21-s + 1.17·23-s + 3.82·25-s − 5.22i·27-s − 8.28i·29-s − 5.65·31-s + ⋯ |
| L(s) = 1 | − 0.624i·3-s − 0.484i·5-s + 0.377·7-s + 0.609·9-s + 1.57i·11-s + 1.74i·13-s − 0.302·15-s + 0.886·17-s + 0.950i·19-s − 0.236i·21-s + 0.244·23-s + 0.765·25-s − 1.00i·27-s − 1.53i·29-s − 1.01·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.75629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.75629\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 1.08iT - 3T^{2} \) |
| 5 | \( 1 + 1.08iT - 5T^{2} \) |
| 11 | \( 1 - 5.22iT - 11T^{2} \) |
| 13 | \( 1 - 6.30iT - 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 - 4.14iT - 19T^{2} \) |
| 23 | \( 1 - 1.17T + 23T^{2} \) |
| 29 | \( 1 + 8.28iT - 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 + 2.16iT - 37T^{2} \) |
| 41 | \( 1 - 7.65T + 41T^{2} \) |
| 43 | \( 1 + 5.22iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 4.32iT - 53T^{2} \) |
| 59 | \( 1 - 6.30iT - 59T^{2} \) |
| 61 | \( 1 - 7.20iT - 61T^{2} \) |
| 67 | \( 1 + 7.39iT - 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 - 0.343T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 - 5.41iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−9.849156726064147755352925851504, −9.454519281498414674273530039103, −8.327579448605597052140085334912, −7.40153550717709541142597127125, −6.96465473286291381651267397979, −5.82313554600301867329904951194, −4.60944708084158395055226544674, −4.07102037604093919479035763842, −2.14023594186934743079211959290, −1.42026803324858544242387101314,
0.996958915264765278336784276167, 3.00978941287447457524284359054, 3.49319425413048393534753167581, 4.96338075279898886706201487179, 5.54211991132827173148920303965, 6.69790138617319049450406610339, 7.66962862779999371154920004877, 8.442279264226277305484952830927, 9.337886146404264188022812620936, 10.29477962997652369652600853986
