| L(s) = 1 | − 3-s − 3.83·5-s − 3.19·7-s + 9-s − 3.48·11-s + 5.48·13-s + 3.83·15-s − 7.32·17-s + 3.48·19-s + 3.19·21-s + 4.05·23-s + 9.68·25-s − 27-s + 7.83·29-s − 6.97·31-s + 3.48·33-s + 12.2·35-s − 37-s − 5.48·39-s + 2.58·41-s + 4·43-s − 3.83·45-s + 6.39·47-s + 3.21·49-s + 7.32·51-s + 11.8·53-s + 13.3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.71·5-s − 1.20·7-s + 0.333·9-s − 1.05·11-s + 1.52·13-s + 0.989·15-s − 1.77·17-s + 0.800·19-s + 0.697·21-s + 0.844·23-s + 1.93·25-s − 0.192·27-s + 1.45·29-s − 1.25·31-s + 0.607·33-s + 2.07·35-s − 0.164·37-s − 0.878·39-s + 0.403·41-s + 0.609·43-s − 0.571·45-s + 0.932·47-s + 0.459·49-s + 1.02·51-s + 1.63·53-s + 1.80·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5638186196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5638186196\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 + 3.83T + 5T^{2} \) |
| 7 | \( 1 + 3.19T + 7T^{2} \) |
| 11 | \( 1 + 3.48T + 11T^{2} \) |
| 13 | \( 1 - 5.48T + 13T^{2} \) |
| 17 | \( 1 + 7.32T + 17T^{2} \) |
| 19 | \( 1 - 3.48T + 19T^{2} \) |
| 23 | \( 1 - 4.05T + 23T^{2} \) |
| 29 | \( 1 - 7.83T + 29T^{2} \) |
| 31 | \( 1 + 6.97T + 31T^{2} \) |
| 41 | \( 1 - 2.58T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 6.39T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 - 6.58T + 61T^{2} \) |
| 67 | \( 1 + 3.37T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 5.27T + 79T^{2} \) |
| 83 | \( 1 - 9.88T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 + 1.66T + 97T^{2} \) |
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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
−10.48256426089397231946001056785, −9.107604262707099750743768015660, −8.498727339090849702076884371993, −7.42237357197326266088932587147, −6.81613009099625310220989670515, −5.85989860378273583054135043699, −4.65274396674845110193384518800, −3.79340636959595206212594506603, −2.91029178589123099822265764447, −0.59483265547739645620447074888,
0.59483265547739645620447074888, 2.91029178589123099822265764447, 3.79340636959595206212594506603, 4.65274396674845110193384518800, 5.85989860378273583054135043699, 6.81613009099625310220989670515, 7.42237357197326266088932587147, 8.498727339090849702076884371993, 9.107604262707099750743768015660, 10.48256426089397231946001056785
