L(s) = 1 | + 2-s + 2.70·3-s + 4-s + 5-s + 2.70·6-s + 0.508·7-s + 8-s + 4.34·9-s + 10-s − 0.524·11-s + 2.70·12-s + 6.69·13-s + 0.508·14-s + 2.70·15-s + 16-s − 3.24·17-s + 4.34·18-s + 0.166·19-s + 20-s + 1.37·21-s − 0.524·22-s + 1.07·23-s + 2.70·24-s + 25-s + 6.69·26-s + 3.63·27-s + 0.508·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.56·3-s + 0.5·4-s + 0.447·5-s + 1.10·6-s + 0.192·7-s + 0.353·8-s + 1.44·9-s + 0.316·10-s − 0.158·11-s + 0.782·12-s + 1.85·13-s + 0.136·14-s + 0.699·15-s + 0.250·16-s − 0.787·17-s + 1.02·18-s + 0.0381·19-s + 0.223·20-s + 0.300·21-s − 0.111·22-s + 0.223·23-s + 0.553·24-s + 0.200·25-s + 1.31·26-s + 0.699·27-s + 0.0961·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.369256224\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.369256224\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 - 2.70T + 3T^{2} \) |
| 7 | \( 1 - 0.508T + 7T^{2} \) |
| 11 | \( 1 + 0.524T + 11T^{2} \) |
| 13 | \( 1 - 6.69T + 13T^{2} \) |
| 17 | \( 1 + 3.24T + 17T^{2} \) |
| 19 | \( 1 - 0.166T + 19T^{2} \) |
| 23 | \( 1 - 1.07T + 23T^{2} \) |
| 29 | \( 1 - 2.11T + 29T^{2} \) |
| 31 | \( 1 + 7.15T + 31T^{2} \) |
| 37 | \( 1 + 6.51T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 3.01T + 43T^{2} \) |
| 47 | \( 1 + 7.56T + 47T^{2} \) |
| 53 | \( 1 - 4.08T + 53T^{2} \) |
| 59 | \( 1 + 1.50T + 59T^{2} \) |
| 61 | \( 1 - 7.63T + 61T^{2} \) |
| 67 | \( 1 + 9.42T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 3.63T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 6.44T + 83T^{2} \) |
| 89 | \( 1 + 4.50T + 89T^{2} \) |
| 97 | \( 1 - 6.30T + 97T^{2} \) |
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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
−8.499195018051288566610545749313, −7.83083969779918678128195871767, −6.96908215382445121181440954273, −6.26758202442051018120889338074, −5.43395250924615547989254225974, −4.43253487397081338485045750439, −3.67624109117347050228537789564, −3.10269738366139222159990381373, −2.14544223238125580914865306712, −1.43263791609823524805724063243,
1.43263791609823524805724063243, 2.14544223238125580914865306712, 3.10269738366139222159990381373, 3.67624109117347050228537789564, 4.43253487397081338485045750439, 5.43395250924615547989254225974, 6.26758202442051018120889338074, 6.96908215382445121181440954273, 7.83083969779918678128195871767, 8.499195018051288566610545749313