| L(s) = 1 | + 2.43·2-s + 3.91·4-s − 3.94·5-s + 4.49·7-s + 4.66·8-s − 9.60·10-s + 3.37·11-s + 4.18·13-s + 10.9·14-s + 3.50·16-s − 4.49·17-s − 2.81·19-s − 15.4·20-s + 8.19·22-s + 0.958·23-s + 10.5·25-s + 10.1·26-s + 17.5·28-s + 6.61·29-s + 2.18·31-s − 0.799·32-s − 10.9·34-s − 17.7·35-s + 1.74·37-s − 6.84·38-s − 18.4·40-s + 3.20·41-s + ⋯ |
| L(s) = 1 | + 1.71·2-s + 1.95·4-s − 1.76·5-s + 1.69·7-s + 1.64·8-s − 3.03·10-s + 1.01·11-s + 1.16·13-s + 2.91·14-s + 0.875·16-s − 1.09·17-s − 0.645·19-s − 3.45·20-s + 1.74·22-s + 0.199·23-s + 2.11·25-s + 1.99·26-s + 3.32·28-s + 1.22·29-s + 0.392·31-s − 0.141·32-s − 1.87·34-s − 2.99·35-s + 0.286·37-s − 1.11·38-s − 2.91·40-s + 0.500·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3447 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3447 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.253490099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.253490099\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 - 2.43T + 2T^{2} \) |
| 5 | \( 1 + 3.94T + 5T^{2} \) |
| 7 | \( 1 - 4.49T + 7T^{2} \) |
| 11 | \( 1 - 3.37T + 11T^{2} \) |
| 13 | \( 1 - 4.18T + 13T^{2} \) |
| 17 | \( 1 + 4.49T + 17T^{2} \) |
| 19 | \( 1 + 2.81T + 19T^{2} \) |
| 23 | \( 1 - 0.958T + 23T^{2} \) |
| 29 | \( 1 - 6.61T + 29T^{2} \) |
| 31 | \( 1 - 2.18T + 31T^{2} \) |
| 37 | \( 1 - 1.74T + 37T^{2} \) |
| 41 | \( 1 - 3.20T + 41T^{2} \) |
| 43 | \( 1 + 7.51T + 43T^{2} \) |
| 47 | \( 1 + 4.42T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 - 4.61T + 61T^{2} \) |
| 67 | \( 1 - 4.61T + 67T^{2} \) |
| 71 | \( 1 - 8.03T + 71T^{2} \) |
| 73 | \( 1 - 7.58T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 - 2.06T + 83T^{2} \) |
| 89 | \( 1 - 5.89T + 89T^{2} \) |
| 97 | \( 1 + 5.27T + 97T^{2} \) |
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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
−8.358784742116147013356336007833, −7.80974077487345106184874671468, −6.77537636440135648211876377469, −6.43879371647720009906830030738, −5.16335272800132482383381928231, −4.56226402915232059339036218663, −4.08215583210450140488308753836, −3.54359572115838831217171726505, −2.33648866442928217276074614344, −1.15013895606966687911071297530,
1.15013895606966687911071297530, 2.33648866442928217276074614344, 3.54359572115838831217171726505, 4.08215583210450140488308753836, 4.56226402915232059339036218663, 5.16335272800132482383381928231, 6.43879371647720009906830030738, 6.77537636440135648211876377469, 7.80974077487345106184874671468, 8.358784742116147013356336007833
