L(s) = 1 | − 2.72·2-s − 1.34·3-s + 5.42·4-s − 2.18·5-s + 3.66·6-s − 9.33·8-s − 1.18·9-s + 5.96·10-s + 1.04·11-s − 7.30·12-s + 2.94·15-s + 14.5·16-s − 5.29·17-s + 3.23·18-s − 0.756·19-s − 11.8·20-s − 2.85·22-s + 0.653·23-s + 12.5·24-s − 0.216·25-s + 5.63·27-s − 3.10·29-s − 8.02·30-s − 1.02·31-s − 21.1·32-s − 1.41·33-s + 14.4·34-s + ⋯ |
L(s) = 1 | − 1.92·2-s − 0.777·3-s + 2.71·4-s − 0.978·5-s + 1.49·6-s − 3.30·8-s − 0.395·9-s + 1.88·10-s + 0.316·11-s − 2.10·12-s + 0.760·15-s + 3.64·16-s − 1.28·17-s + 0.762·18-s − 0.173·19-s − 2.65·20-s − 0.609·22-s + 0.136·23-s + 2.56·24-s − 0.0432·25-s + 1.08·27-s − 0.576·29-s − 1.46·30-s − 0.184·31-s − 3.73·32-s − 0.245·33-s + 2.47·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 2.72T + 2T^{2} \) |
| 3 | \( 1 + 1.34T + 3T^{2} \) |
| 5 | \( 1 + 2.18T + 5T^{2} \) |
| 11 | \( 1 - 1.04T + 11T^{2} \) |
| 17 | \( 1 + 5.29T + 17T^{2} \) |
| 19 | \( 1 + 0.756T + 19T^{2} \) |
| 23 | \( 1 - 0.653T + 23T^{2} \) |
| 29 | \( 1 + 3.10T + 29T^{2} \) |
| 31 | \( 1 + 1.02T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 + 7.32T + 41T^{2} \) |
| 43 | \( 1 - 0.887T + 43T^{2} \) |
| 47 | \( 1 + 2.33T + 47T^{2} \) |
| 53 | \( 1 - 4.88T + 53T^{2} \) |
| 59 | \( 1 - 1.04T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 4.47T + 67T^{2} \) |
| 71 | \( 1 - 6.60T + 71T^{2} \) |
| 73 | \( 1 - 8.28T + 73T^{2} \) |
| 79 | \( 1 - 2.14T + 79T^{2} \) |
| 83 | \( 1 - 6.66T + 83T^{2} \) |
| 89 | \( 1 - 5.76T + 89T^{2} \) |
| 97 | \( 1 - 2.88T + 97T^{2} \) |
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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
−7.74019460740866161490806390520, −6.86513425858622458974280133722, −6.44071785646805459973215309541, −5.76628819734679932872917788752, −4.71252340065477595832335883337, −3.70576584173178193877144449576, −2.75726408968915537855891981205, −1.88663092786401848700713472869, −0.73532680424079142632935069130, 0,
0.73532680424079142632935069130, 1.88663092786401848700713472869, 2.75726408968915537855891981205, 3.70576584173178193877144449576, 4.71252340065477595832335883337, 5.76628819734679932872917788752, 6.44071785646805459973215309541, 6.86513425858622458974280133722, 7.74019460740866161490806390520