| L(s) = 1 | + 2.82·3-s − 5-s + 2.82·7-s + 5.00·9-s − 5.65·11-s + 2·13-s − 2.82·15-s + 2·17-s + 8.00·21-s − 2.82·23-s + 25-s + 5.65·27-s − 6·29-s − 5.65·31-s − 16.0·33-s − 2.82·35-s + 10·37-s + 5.65·39-s + 2·41-s − 8.48·43-s − 5.00·45-s + 2.82·47-s + 1.00·49-s + 5.65·51-s − 6·53-s + 5.65·55-s + 11.3·59-s + ⋯ |
| L(s) = 1 | + 1.63·3-s − 0.447·5-s + 1.06·7-s + 1.66·9-s − 1.70·11-s + 0.554·13-s − 0.730·15-s + 0.485·17-s + 1.74·21-s − 0.589·23-s + 0.200·25-s + 1.08·27-s − 1.11·29-s − 1.01·31-s − 2.78·33-s − 0.478·35-s + 1.64·37-s + 0.905·39-s + 0.312·41-s − 1.29·43-s − 0.745·45-s + 0.412·47-s + 0.142·49-s + 0.792·51-s − 0.824·53-s + 0.762·55-s + 1.47·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.092845398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.092845398\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 3 | \( 1 - 2.82T + 3T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 2.82T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 2.82T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−11.53817667191451373632568628568, −10.63652798535804361642446776952, −9.626869907218388433642260600245, −8.519721980105615848195832680186, −7.943670102081038157537312274234, −7.42867144191638870823373674443, −5.51245569181223824742894392200, −4.27226961306955603738878899232, −3.12925247737475439352160521810, −1.95006760437674357452119030431,
1.95006760437674357452119030431, 3.12925247737475439352160521810, 4.27226961306955603738878899232, 5.51245569181223824742894392200, 7.42867144191638870823373674443, 7.943670102081038157537312274234, 8.519721980105615848195832680186, 9.626869907218388433642260600245, 10.63652798535804361642446776952, 11.53817667191451373632568628568
