| L(s) = 1 | + 2-s + 2.61·3-s + 4-s − 5-s + 2.61·6-s + 7-s + 8-s + 3.85·9-s − 10-s + 2.61·12-s + 2·13-s + 14-s − 2.61·15-s + 16-s − 1.61·17-s + 3.85·18-s + 6.85·19-s − 20-s + 2.61·21-s + 6·23-s + 2.61·24-s + 25-s + 2·26-s + 2.23·27-s + 28-s + 3.23·29-s − 2.61·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.51·3-s + 0.5·4-s − 0.447·5-s + 1.06·6-s + 0.377·7-s + 0.353·8-s + 1.28·9-s − 0.316·10-s + 0.755·12-s + 0.554·13-s + 0.267·14-s − 0.675·15-s + 0.250·16-s − 0.392·17-s + 0.908·18-s + 1.57·19-s − 0.223·20-s + 0.571·21-s + 1.25·23-s + 0.534·24-s + 0.200·25-s + 0.392·26-s + 0.430·27-s + 0.188·28-s + 0.600·29-s − 0.477·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.385163308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.385163308\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2.61T + 3T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 1.61T + 17T^{2} \) |
| 19 | \( 1 - 6.85T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 3.23T + 29T^{2} \) |
| 31 | \( 1 + 1.23T + 31T^{2} \) |
| 37 | \( 1 + 6.47T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 1.85T + 43T^{2} \) |
| 47 | \( 1 - 9.23T + 47T^{2} \) |
| 53 | \( 1 - 1.23T + 53T^{2} \) |
| 59 | \( 1 + 7.61T + 59T^{2} \) |
| 61 | \( 1 + 3.52T + 61T^{2} \) |
| 67 | \( 1 - 6.09T + 67T^{2} \) |
| 71 | \( 1 + 9.70T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 + 8.47T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 9.56T + 97T^{2} \) |
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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
−7.74515238481860124047935888291, −7.22583259874961772267331316318, −6.61251981423924699059184993406, −5.49354971531854480003293368294, −4.88775267676840082692348819403, −4.06518134710107328686888570203, −3.34833684309992433015003167530, −2.97733729146167180574103667338, −2.00348214207417670392999231143, −1.11082580664517403773910388091,
1.11082580664517403773910388091, 2.00348214207417670392999231143, 2.97733729146167180574103667338, 3.34833684309992433015003167530, 4.06518134710107328686888570203, 4.88775267676840082692348819403, 5.49354971531854480003293368294, 6.61251981423924699059184993406, 7.22583259874961772267331316318, 7.74515238481860124047935888291
