| L(s) = 1 | + 2.39·2-s + 3.73·4-s − 5-s − 1.96·7-s + 4.15·8-s − 2.39·10-s − 2.92·13-s − 4.71·14-s + 2.48·16-s + 6.62·17-s + 3.56·19-s − 3.73·20-s + 2.78·23-s + 25-s − 6.99·26-s − 7.35·28-s + 3.51·29-s + 8.11·31-s − 2.36·32-s + 15.8·34-s + 1.96·35-s + 4.42·37-s + 8.52·38-s − 4.15·40-s − 7.07·41-s + 10.8·43-s + 6.67·46-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 1.86·4-s − 0.447·5-s − 0.744·7-s + 1.46·8-s − 0.757·10-s − 0.809·13-s − 1.26·14-s + 0.620·16-s + 1.60·17-s + 0.816·19-s − 0.835·20-s + 0.581·23-s + 0.200·25-s − 1.37·26-s − 1.39·28-s + 0.653·29-s + 1.45·31-s − 0.418·32-s + 2.71·34-s + 0.332·35-s + 0.728·37-s + 1.38·38-s − 0.657·40-s − 1.10·41-s + 1.66·43-s + 0.984·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.921752124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.921752124\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.39T + 2T^{2} \) |
| 7 | \( 1 + 1.96T + 7T^{2} \) |
| 13 | \( 1 + 2.92T + 13T^{2} \) |
| 17 | \( 1 - 6.62T + 17T^{2} \) |
| 19 | \( 1 - 3.56T + 19T^{2} \) |
| 23 | \( 1 - 2.78T + 23T^{2} \) |
| 29 | \( 1 - 3.51T + 29T^{2} \) |
| 31 | \( 1 - 8.11T + 31T^{2} \) |
| 37 | \( 1 - 4.42T + 37T^{2} \) |
| 41 | \( 1 + 7.07T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 0.0271T + 47T^{2} \) |
| 53 | \( 1 - 4.10T + 53T^{2} \) |
| 59 | \( 1 + 9.51T + 59T^{2} \) |
| 61 | \( 1 + 3.10T + 61T^{2} \) |
| 67 | \( 1 - 1.58T + 67T^{2} \) |
| 71 | \( 1 + 6.05T + 71T^{2} \) |
| 73 | \( 1 - 2.37T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 - 5.04T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−7.65990553712241166511087355001, −7.41862657215368564693095219986, −6.40869398092638960512450600244, −5.99141318865899045263823605303, −5.00947875029820967593475525836, −4.68189813895199597888233476605, −3.56724505269652833292832951755, −3.17410202346306568293244455385, −2.43485255377419154675953850030, −0.931029391076415203797513315672,
0.931029391076415203797513315672, 2.43485255377419154675953850030, 3.17410202346306568293244455385, 3.56724505269652833292832951755, 4.68189813895199597888233476605, 5.00947875029820967593475525836, 5.99141318865899045263823605303, 6.40869398092638960512450600244, 7.41862657215368564693095219986, 7.65990553712241166511087355001
