L(s) = 1 | + 2-s + 4-s + 2.92·5-s − 3.28·7-s + 8-s + 2.92·10-s − 0.882·11-s − 2.93·13-s − 3.28·14-s + 16-s + 2.03·17-s − 2.83·19-s + 2.92·20-s − 0.882·22-s − 7.21·23-s + 3.57·25-s − 2.93·26-s − 3.28·28-s − 1.57·29-s + 3.45·31-s + 32-s + 2.03·34-s − 9.61·35-s + 0.228·37-s − 2.83·38-s + 2.92·40-s + 7.96·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.30·5-s − 1.24·7-s + 0.353·8-s + 0.925·10-s − 0.266·11-s − 0.813·13-s − 0.877·14-s + 0.250·16-s + 0.492·17-s − 0.649·19-s + 0.654·20-s − 0.188·22-s − 1.50·23-s + 0.714·25-s − 0.575·26-s − 0.620·28-s − 0.293·29-s + 0.619·31-s + 0.176·32-s + 0.348·34-s − 1.62·35-s + 0.0374·37-s − 0.459·38-s + 0.462·40-s + 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
good | 5 | \( 1 - 2.92T + 5T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 11 | \( 1 + 0.882T + 11T^{2} \) |
| 13 | \( 1 + 2.93T + 13T^{2} \) |
| 17 | \( 1 - 2.03T + 17T^{2} \) |
| 19 | \( 1 + 2.83T + 19T^{2} \) |
| 23 | \( 1 + 7.21T + 23T^{2} \) |
| 29 | \( 1 + 1.57T + 29T^{2} \) |
| 31 | \( 1 - 3.45T + 31T^{2} \) |
| 37 | \( 1 - 0.228T + 37T^{2} \) |
| 41 | \( 1 - 7.96T + 41T^{2} \) |
| 43 | \( 1 - 1.77T + 43T^{2} \) |
| 47 | \( 1 + 4.83T + 47T^{2} \) |
| 53 | \( 1 + 6.05T + 53T^{2} \) |
| 59 | \( 1 + 14.7T + 59T^{2} \) |
| 61 | \( 1 - 8.51T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 3.74T + 71T^{2} \) |
| 73 | \( 1 - 5.46T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 8.73T + 83T^{2} \) |
| 89 | \( 1 + 18.0T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−7.33224390120740772436019247873, −6.43978123260993086572642675482, −6.10268453980106062913880979932, −5.57290740720639379958750953050, −4.70472280419750077817659351826, −3.92670179047158059852766909103, −2.95679218742173203016473706123, −2.44393772393431251360062665533, −1.57477545734102028870811201116, 0,
1.57477545734102028870811201116, 2.44393772393431251360062665533, 2.95679218742173203016473706123, 3.92670179047158059852766909103, 4.70472280419750077817659351826, 5.57290740720639379958750953050, 6.10268453980106062913880979932, 6.43978123260993086572642675482, 7.33224390120740772436019247873