| L(s) = 1 | − 1.79·7-s + 4.83·11-s − 1.79·13-s + 3.10·17-s − 5.58·19-s + 3.10·23-s − 3.46·29-s − 2.58·31-s + 10.3·37-s + 1.37·41-s − 1.79·43-s + 8.66·47-s − 3.79·49-s − 3.10·53-s + 0.361·59-s − 3.37·61-s + 5·67-s − 7.93·71-s + 5·73-s − 8.66·77-s − 4·79-s + 14.8·83-s − 2.74·89-s + 3.20·91-s + 17.1·97-s + 7.28·101-s + 10.3·103-s + ⋯ |
| L(s) = 1 | − 0.677·7-s + 1.45·11-s − 0.496·13-s + 0.752·17-s − 1.28·19-s + 0.646·23-s − 0.643·29-s − 0.463·31-s + 1.70·37-s + 0.214·41-s − 0.273·43-s + 1.26·47-s − 0.541·49-s − 0.426·53-s + 0.0470·59-s − 0.431·61-s + 0.610·67-s − 0.941·71-s + 0.585·73-s − 0.986·77-s − 0.450·79-s + 1.63·83-s − 0.290·89-s + 0.336·91-s + 1.74·97-s + 0.725·101-s + 1.02·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.854139480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.854139480\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 1.79T + 7T^{2} \) |
| 11 | \( 1 - 4.83T + 11T^{2} \) |
| 13 | \( 1 + 1.79T + 13T^{2} \) |
| 17 | \( 1 - 3.10T + 17T^{2} \) |
| 19 | \( 1 + 5.58T + 19T^{2} \) |
| 23 | \( 1 - 3.10T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 2.58T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 - 1.37T + 41T^{2} \) |
| 43 | \( 1 + 1.79T + 43T^{2} \) |
| 47 | \( 1 - 8.66T + 47T^{2} \) |
| 53 | \( 1 + 3.10T + 53T^{2} \) |
| 59 | \( 1 - 0.361T + 59T^{2} \) |
| 61 | \( 1 + 3.37T + 61T^{2} \) |
| 67 | \( 1 - 5T + 67T^{2} \) |
| 71 | \( 1 + 7.93T + 71T^{2} \) |
| 73 | \( 1 - 5T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 + 2.74T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−7.72673598343020700278848437987, −7.10613542992169646257707962970, −6.35538545635633428361532092569, −5.99013707360036739143876947693, −4.97689017198815235249806607235, −4.17083961694389233292343297572, −3.59041061018980204964332271305, −2.71111766245530919875862693057, −1.74894485648376808100540525275, −0.67963789628536850746450384065,
0.67963789628536850746450384065, 1.74894485648376808100540525275, 2.71111766245530919875862693057, 3.59041061018980204964332271305, 4.17083961694389233292343297572, 4.97689017198815235249806607235, 5.99013707360036739143876947693, 6.35538545635633428361532092569, 7.10613542992169646257707962970, 7.72673598343020700278848437987
