| L(s) = 1 | + 2-s + 2.22·3-s + 4-s + 1.86·5-s + 2.22·6-s − 2.90·7-s + 8-s + 1.95·9-s + 1.86·10-s − 5.12·11-s + 2.22·12-s − 6.02·13-s − 2.90·14-s + 4.14·15-s + 16-s + 4.14·17-s + 1.95·18-s − 4.74·19-s + 1.86·20-s − 6.46·21-s − 5.12·22-s − 5.93·23-s + 2.22·24-s − 1.53·25-s − 6.02·26-s − 2.32·27-s − 2.90·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.28·3-s + 0.5·4-s + 0.832·5-s + 0.908·6-s − 1.09·7-s + 0.353·8-s + 0.651·9-s + 0.588·10-s − 1.54·11-s + 0.642·12-s − 1.67·13-s − 0.776·14-s + 1.06·15-s + 0.250·16-s + 1.00·17-s + 0.460·18-s − 1.08·19-s + 0.416·20-s − 1.41·21-s − 1.09·22-s − 1.23·23-s + 0.454·24-s − 0.307·25-s − 1.18·26-s − 0.447·27-s − 0.548·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 - 2.22T + 3T^{2} \) |
| 5 | \( 1 - 1.86T + 5T^{2} \) |
| 7 | \( 1 + 2.90T + 7T^{2} \) |
| 11 | \( 1 + 5.12T + 11T^{2} \) |
| 13 | \( 1 + 6.02T + 13T^{2} \) |
| 17 | \( 1 - 4.14T + 17T^{2} \) |
| 19 | \( 1 + 4.74T + 19T^{2} \) |
| 23 | \( 1 + 5.93T + 23T^{2} \) |
| 29 | \( 1 - 2.07T + 29T^{2} \) |
| 37 | \( 1 + 7.19T + 37T^{2} \) |
| 41 | \( 1 - 4.62T + 41T^{2} \) |
| 43 | \( 1 - 6.37T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 2.76T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 + 5.90T + 71T^{2} \) |
| 73 | \( 1 - 8.57T + 73T^{2} \) |
| 79 | \( 1 - 2.29T + 79T^{2} \) |
| 83 | \( 1 - 3.20T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
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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
−7.77685865676033426018691195789, −7.10434524572828586024427830949, −6.14429184702451484889341768356, −5.62159456214448099988923243554, −4.81330859049761393264056475464, −3.87344425810170409961938258119, −3.02702326217990916045134409679, −2.48291478458630670523465794277, −2.03652105096598294753938332016, 0,
2.03652105096598294753938332016, 2.48291478458630670523465794277, 3.02702326217990916045134409679, 3.87344425810170409961938258119, 4.81330859049761393264056475464, 5.62159456214448099988923243554, 6.14429184702451484889341768356, 7.10434524572828586024427830949, 7.77685865676033426018691195789
