| L(s) = 1 | + 3-s + 3.22·5-s + 0.874·7-s + 9-s − 11-s + 13-s + 3.22·15-s − 1.16·17-s + 0.874·19-s + 0.874·21-s − 8.84·23-s + 5.39·25-s + 27-s + 1.65·29-s + 1.22·31-s − 33-s + 2.82·35-s − 0.820·37-s + 39-s + 11.6·41-s − 4.91·43-s + 3.22·45-s + 12.4·47-s − 6.23·49-s − 1.16·51-s + 4.44·53-s − 3.22·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.44·5-s + 0.330·7-s + 0.333·9-s − 0.301·11-s + 0.277·13-s + 0.832·15-s − 0.283·17-s + 0.200·19-s + 0.190·21-s − 1.84·23-s + 1.07·25-s + 0.192·27-s + 0.306·29-s + 0.219·31-s − 0.174·33-s + 0.476·35-s − 0.134·37-s + 0.160·39-s + 1.82·41-s − 0.750·43-s + 0.480·45-s + 1.81·47-s − 0.890·49-s − 0.163·51-s + 0.610·53-s − 0.434·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.650531210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.650531210\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 3.22T + 5T^{2} \) |
| 7 | \( 1 - 0.874T + 7T^{2} \) |
| 17 | \( 1 + 1.16T + 17T^{2} \) |
| 19 | \( 1 - 0.874T + 19T^{2} \) |
| 23 | \( 1 + 8.84T + 23T^{2} \) |
| 29 | \( 1 - 1.65T + 29T^{2} \) |
| 31 | \( 1 - 1.22T + 31T^{2} \) |
| 37 | \( 1 + 0.820T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 4.91T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 4.44T + 53T^{2} \) |
| 59 | \( 1 - 6.44T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 7.71T + 71T^{2} \) |
| 73 | \( 1 - 5.57T + 73T^{2} \) |
| 79 | \( 1 + 2.47T + 79T^{2} \) |
| 83 | \( 1 + 16.9T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−8.072484760665017311286289363363, −7.31324386597739071875386447539, −6.45243711107950089474855917265, −5.87465897277856314855016259382, −5.25160685033564417152365619378, −4.34217731999479874058397999208, −3.55704810542718521253414570612, −2.32607864528845087923301960922, −2.16036146637275080956616887032, −0.977058343489906396051428939199,
0.977058343489906396051428939199, 2.16036146637275080956616887032, 2.32607864528845087923301960922, 3.55704810542718521253414570612, 4.34217731999479874058397999208, 5.25160685033564417152365619378, 5.87465897277856314855016259382, 6.45243711107950089474855917265, 7.31324386597739071875386447539, 8.072484760665017311286289363363
