L(s) = 1 | + 2.67·2-s − 3-s + 5.16·4-s − 1.98·5-s − 2.67·6-s + 0.0781·7-s + 8.45·8-s + 9-s − 5.31·10-s + 3.41·11-s − 5.16·12-s − 13-s + 0.209·14-s + 1.98·15-s + 12.3·16-s + 4.69·17-s + 2.67·18-s + 1.61·19-s − 10.2·20-s − 0.0781·21-s + 9.12·22-s − 1.87·23-s − 8.45·24-s − 1.05·25-s − 2.67·26-s − 27-s + 0.403·28-s + ⋯ |
L(s) = 1 | + 1.89·2-s − 0.577·3-s + 2.58·4-s − 0.888·5-s − 1.09·6-s + 0.0295·7-s + 2.99·8-s + 0.333·9-s − 1.68·10-s + 1.02·11-s − 1.48·12-s − 0.277·13-s + 0.0559·14-s + 0.513·15-s + 3.07·16-s + 1.13·17-s + 0.630·18-s + 0.370·19-s − 2.29·20-s − 0.0170·21-s + 1.94·22-s − 0.391·23-s − 1.72·24-s − 0.210·25-s − 0.524·26-s − 0.192·27-s + 0.0762·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.324111352\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.324111352\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 5 | \( 1 + 1.98T + 5T^{2} \) |
| 7 | \( 1 - 0.0781T + 7T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 17 | \( 1 - 4.69T + 17T^{2} \) |
| 19 | \( 1 - 1.61T + 19T^{2} \) |
| 23 | \( 1 + 1.87T + 23T^{2} \) |
| 29 | \( 1 + 2.33T + 29T^{2} \) |
| 31 | \( 1 + 2.02T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 - 3.38T + 41T^{2} \) |
| 43 | \( 1 - 5.80T + 43T^{2} \) |
| 47 | \( 1 - 1.16T + 47T^{2} \) |
| 53 | \( 1 - 4.15T + 53T^{2} \) |
| 59 | \( 1 + 1.92T + 59T^{2} \) |
| 61 | \( 1 - 0.0325T + 61T^{2} \) |
| 67 | \( 1 - 7.23T + 67T^{2} \) |
| 71 | \( 1 - 7.99T + 71T^{2} \) |
| 73 | \( 1 + 2.29T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 2.42T + 83T^{2} \) |
| 89 | \( 1 - 4.51T + 89T^{2} \) |
| 97 | \( 1 - 8.99T + 97T^{2} \) |
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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
−7.82372142000567243654048931354, −7.58641854058728778625642816724, −6.66639822416726950780263795456, −6.04602031617631228002995397622, −5.39948959679098030920964521373, −4.59383755960270376185892607509, −3.91062126407335507889531116117, −3.45451457826270264464507885083, −2.31173596529492049203824528880, −1.09118368039380236256635971950,
1.09118368039380236256635971950, 2.31173596529492049203824528880, 3.45451457826270264464507885083, 3.91062126407335507889531116117, 4.59383755960270376185892607509, 5.39948959679098030920964521373, 6.04602031617631228002995397622, 6.66639822416726950780263795456, 7.58641854058728778625642816724, 7.82372142000567243654048931354