| L(s) = 1 | − 2.61·3-s + 4.05·5-s − 3.36·7-s + 3.83·9-s − 11-s − 6.11·13-s − 10.5·15-s − 5.67·17-s + 19-s + 8.78·21-s − 2.75·23-s + 11.4·25-s − 2.17·27-s − 7.22·29-s + 0.528·31-s + 2.61·33-s − 13.6·35-s + 1.47·37-s + 15.9·39-s + 11.6·41-s + 11.6·43-s + 15.5·45-s + 4.10·47-s + 4.29·49-s + 14.8·51-s − 7.48·53-s − 4.05·55-s + ⋯ |
| L(s) = 1 | − 1.50·3-s + 1.81·5-s − 1.27·7-s + 1.27·9-s − 0.301·11-s − 1.69·13-s − 2.73·15-s − 1.37·17-s + 0.229·19-s + 1.91·21-s − 0.573·23-s + 2.28·25-s − 0.418·27-s − 1.34·29-s + 0.0949·31-s + 0.455·33-s − 2.30·35-s + 0.241·37-s + 2.55·39-s + 1.81·41-s + 1.77·43-s + 2.31·45-s + 0.598·47-s + 0.614·49-s + 2.07·51-s − 1.02·53-s − 0.546·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8127633107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8127633107\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 - 4.05T + 5T^{2} \) |
| 7 | \( 1 + 3.36T + 7T^{2} \) |
| 13 | \( 1 + 6.11T + 13T^{2} \) |
| 17 | \( 1 + 5.67T + 17T^{2} \) |
| 23 | \( 1 + 2.75T + 23T^{2} \) |
| 29 | \( 1 + 7.22T + 29T^{2} \) |
| 31 | \( 1 - 0.528T + 31T^{2} \) |
| 37 | \( 1 - 1.47T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 - 4.10T + 47T^{2} \) |
| 53 | \( 1 + 7.48T + 53T^{2} \) |
| 59 | \( 1 - 9.90T + 59T^{2} \) |
| 61 | \( 1 + 5.52T + 61T^{2} \) |
| 67 | \( 1 - 1.71T + 67T^{2} \) |
| 71 | \( 1 - 3.95T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 3.56T + 79T^{2} \) |
| 83 | \( 1 - 2.35T + 83T^{2} \) |
| 89 | \( 1 + 2.30T + 89T^{2} \) |
| 97 | \( 1 - 0.409T + 97T^{2} \) |
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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
−9.100313453263411424095826605531, −7.50363090398919503916657911581, −6.82480396310200165660245758539, −6.19209187394376935192347700049, −5.74502880001247796929359690774, −5.10681609166522260706536325293, −4.26443342656077459549526888327, −2.68994442126110925483593537441, −2.10079225969882716257356502154, −0.54808372036542595179463430954,
0.54808372036542595179463430954, 2.10079225969882716257356502154, 2.68994442126110925483593537441, 4.26443342656077459549526888327, 5.10681609166522260706536325293, 5.74502880001247796929359690774, 6.19209187394376935192347700049, 6.82480396310200165660245758539, 7.50363090398919503916657911581, 9.100313453263411424095826605531
