| L(s) = 1 | − 2-s + 1.84·3-s + 4-s + 1.37·5-s − 1.84·6-s − 4.01·7-s − 8-s + 0.412·9-s − 1.37·10-s + 3.91·11-s + 1.84·12-s + 4.20·13-s + 4.01·14-s + 2.54·15-s + 16-s + 2.78·17-s − 0.412·18-s + 0.849·19-s + 1.37·20-s − 7.42·21-s − 3.91·22-s + 23-s − 1.84·24-s − 3.10·25-s − 4.20·26-s − 4.77·27-s − 4.01·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.06·3-s + 0.5·4-s + 0.614·5-s − 0.754·6-s − 1.51·7-s − 0.353·8-s + 0.137·9-s − 0.434·10-s + 1.18·11-s + 0.533·12-s + 1.16·13-s + 1.07·14-s + 0.655·15-s + 0.250·16-s + 0.675·17-s − 0.0973·18-s + 0.194·19-s + 0.307·20-s − 1.61·21-s − 0.835·22-s + 0.208·23-s − 0.377·24-s − 0.621·25-s − 0.823·26-s − 0.919·27-s − 0.759·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.218319361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.218319361\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 1.84T + 3T^{2} \) |
| 5 | \( 1 - 1.37T + 5T^{2} \) |
| 7 | \( 1 + 4.01T + 7T^{2} \) |
| 11 | \( 1 - 3.91T + 11T^{2} \) |
| 13 | \( 1 - 4.20T + 13T^{2} \) |
| 17 | \( 1 - 2.78T + 17T^{2} \) |
| 19 | \( 1 - 0.849T + 19T^{2} \) |
| 29 | \( 1 + 0.467T + 29T^{2} \) |
| 31 | \( 1 + 4.92T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 5.35T + 41T^{2} \) |
| 43 | \( 1 - 0.879T + 43T^{2} \) |
| 47 | \( 1 - 5.13T + 47T^{2} \) |
| 53 | \( 1 - 0.0180T + 53T^{2} \) |
| 59 | \( 1 - 8.99T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 2.34T + 67T^{2} \) |
| 71 | \( 1 - 5.88T + 71T^{2} \) |
| 73 | \( 1 + 15.5T + 73T^{2} \) |
| 79 | \( 1 + 3.43T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 - 7.62T + 89T^{2} \) |
| 97 | \( 1 - 2.26T + 97T^{2} \) |
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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
−8.256783788395089948792945826247, −7.47377716341403304666373932610, −6.69918664669394091727013163390, −6.11048630637957673354418837213, −5.59627036386356193854781667007, −3.89607473958443368363468859188, −3.55073135069707722573467351216, −2.75564853687862256102189887894, −1.87408054761074832291962576123, −0.837708726286008450051523406271,
0.837708726286008450051523406271, 1.87408054761074832291962576123, 2.75564853687862256102189887894, 3.55073135069707722573467351216, 3.89607473958443368363468859188, 5.59627036386356193854781667007, 6.11048630637957673354418837213, 6.69918664669394091727013163390, 7.47377716341403304666373932610, 8.256783788395089948792945826247
