| L(s) = 1 | + 1.07·3-s − 0.402·5-s − 7-s − 1.84·9-s + 5.12·11-s − 13-s − 0.432·15-s − 3.40·17-s − 2.40·19-s − 1.07·21-s + 1.49·23-s − 4.83·25-s − 5.20·27-s + 10.0·29-s + 9.13·31-s + 5.50·33-s + 0.402·35-s + 11.9·37-s − 1.07·39-s + 5.48·41-s + 6.15·43-s + 0.743·45-s + 0.731·47-s + 49-s − 3.65·51-s + 5.14·53-s − 2.06·55-s + ⋯ |
| L(s) = 1 | + 0.620·3-s − 0.180·5-s − 0.377·7-s − 0.615·9-s + 1.54·11-s − 0.277·13-s − 0.111·15-s − 0.826·17-s − 0.551·19-s − 0.234·21-s + 0.312·23-s − 0.967·25-s − 1.00·27-s + 1.85·29-s + 1.64·31-s + 0.958·33-s + 0.0680·35-s + 1.96·37-s − 0.172·39-s + 0.856·41-s + 0.939·43-s + 0.110·45-s + 0.106·47-s + 0.142·49-s − 0.512·51-s + 0.706·53-s − 0.278·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.078663490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.078663490\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 1.07T + 3T^{2} \) |
| 5 | \( 1 + 0.402T + 5T^{2} \) |
| 11 | \( 1 - 5.12T + 11T^{2} \) |
| 17 | \( 1 + 3.40T + 17T^{2} \) |
| 19 | \( 1 + 2.40T + 19T^{2} \) |
| 23 | \( 1 - 1.49T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 - 9.13T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 - 5.48T + 41T^{2} \) |
| 43 | \( 1 - 6.15T + 43T^{2} \) |
| 47 | \( 1 - 0.731T + 47T^{2} \) |
| 53 | \( 1 - 5.14T + 53T^{2} \) |
| 59 | \( 1 + 7.51T + 59T^{2} \) |
| 61 | \( 1 - 3.67T + 61T^{2} \) |
| 67 | \( 1 + 4.82T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 7.46T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 0.310T + 83T^{2} \) |
| 89 | \( 1 - 2.92T + 89T^{2} \) |
| 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
−8.748523214312535371907634008117, −8.186648781581837532782510383328, −7.29620428696892098414868049067, −6.34361128036733655922857737232, −6.04693239344916517843921707317, −4.53804476604863484674795868240, −4.09621451617431268849832872963, −2.99138950100991577093193024718, −2.31123377952615930614175862631, −0.863188837038228021869867072720,
0.863188837038228021869867072720, 2.31123377952615930614175862631, 2.99138950100991577093193024718, 4.09621451617431268849832872963, 4.53804476604863484674795868240, 6.04693239344916517843921707317, 6.34361128036733655922857737232, 7.29620428696892098414868049067, 8.186648781581837532782510383328, 8.748523214312535371907634008117
