| L(s) = 1 | − 0.136·2-s + 2.96·3-s − 1.98·4-s + 3.29·5-s − 0.404·6-s + 0.543·8-s + 5.80·9-s − 0.448·10-s − 0.874·11-s − 5.87·12-s + 9.76·15-s + 3.88·16-s + 5.43·17-s − 0.791·18-s + 1.86·19-s − 6.51·20-s + 0.119·22-s − 6.86·23-s + 1.61·24-s + 5.82·25-s + 8.31·27-s − 4.15·29-s − 1.33·30-s + 9.04·31-s − 1.61·32-s − 2.59·33-s − 0.742·34-s + ⋯ |
| L(s) = 1 | − 0.0964·2-s + 1.71·3-s − 0.990·4-s + 1.47·5-s − 0.165·6-s + 0.192·8-s + 1.93·9-s − 0.141·10-s − 0.263·11-s − 1.69·12-s + 2.52·15-s + 0.972·16-s + 1.31·17-s − 0.186·18-s + 0.428·19-s − 1.45·20-s + 0.0254·22-s − 1.43·23-s + 0.329·24-s + 1.16·25-s + 1.59·27-s − 0.771·29-s − 0.243·30-s + 1.62·31-s − 0.285·32-s − 0.451·33-s − 0.127·34-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
| \(L(1)\) |
\(\approx\) |
\(4.452035240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.452035240\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.136T + 2T^{2} \) |
| 3 | \( 1 - 2.96T + 3T^{2} \) |
| 5 | \( 1 - 3.29T + 5T^{2} \) |
| 11 | \( 1 + 0.874T + 11T^{2} \) |
| 17 | \( 1 - 5.43T + 17T^{2} \) |
| 19 | \( 1 - 1.86T + 19T^{2} \) |
| 23 | \( 1 + 6.86T + 23T^{2} \) |
| 29 | \( 1 + 4.15T + 29T^{2} \) |
| 31 | \( 1 - 9.04T + 31T^{2} \) |
| 37 | \( 1 + 0.719T + 37T^{2} \) |
| 41 | \( 1 + 0.916T + 41T^{2} \) |
| 43 | \( 1 - 5.76T + 43T^{2} \) |
| 47 | \( 1 + 3.51T + 47T^{2} \) |
| 53 | \( 1 - 4.82T + 53T^{2} \) |
| 59 | \( 1 - 4.84T + 59T^{2} \) |
| 61 | \( 1 + 0.968T + 61T^{2} \) |
| 67 | \( 1 + 6.84T + 67T^{2} \) |
| 71 | \( 1 - 6.91T + 71T^{2} \) |
| 73 | \( 1 + 3.29T + 73T^{2} \) |
| 79 | \( 1 - 9.00T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 9.67T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−7.941895963491282635545925085016, −7.51718399018030111323368504541, −6.37468550064503890686131890010, −5.66175559890944228724447412696, −5.00381278815839272122339774896, −4.07502201320899967737757914703, −3.42326468920229988755071554864, −2.63222099941585297848971786803, −1.89767353151229360420476659129, −1.04614534549262474737386557649,
1.04614534549262474737386557649, 1.89767353151229360420476659129, 2.63222099941585297848971786803, 3.42326468920229988755071554864, 4.07502201320899967737757914703, 5.00381278815839272122339774896, 5.66175559890944228724447412696, 6.37468550064503890686131890010, 7.51718399018030111323368504541, 7.941895963491282635545925085016
