L(s) = 1 | + 2.15·2-s − 2.12·3-s + 2.65·4-s + 2.93·5-s − 4.57·6-s − 2.45·7-s + 1.40·8-s + 1.49·9-s + 6.33·10-s + 6.14·11-s − 5.62·12-s − 0.596·13-s − 5.28·14-s − 6.22·15-s − 2.26·16-s + 0.589·17-s + 3.22·18-s + 1.88·19-s + 7.78·20-s + 5.19·21-s + 13.2·22-s + 3.94·23-s − 2.98·24-s + 3.61·25-s − 1.28·26-s + 3.19·27-s − 6.50·28-s + ⋯ |
L(s) = 1 | + 1.52·2-s − 1.22·3-s + 1.32·4-s + 1.31·5-s − 1.86·6-s − 0.926·7-s + 0.498·8-s + 0.498·9-s + 2.00·10-s + 1.85·11-s − 1.62·12-s − 0.165·13-s − 1.41·14-s − 1.60·15-s − 0.566·16-s + 0.142·17-s + 0.760·18-s + 0.432·19-s + 1.74·20-s + 1.13·21-s + 2.82·22-s + 0.822·23-s − 0.609·24-s + 0.722·25-s − 0.252·26-s + 0.613·27-s − 1.22·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.272265133\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.272265133\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 41 | \( 1 \) |
good | 2 | \( 1 - 2.15T + 2T^{2} \) |
| 3 | \( 1 + 2.12T + 3T^{2} \) |
| 5 | \( 1 - 2.93T + 5T^{2} \) |
| 7 | \( 1 + 2.45T + 7T^{2} \) |
| 11 | \( 1 - 6.14T + 11T^{2} \) |
| 13 | \( 1 + 0.596T + 13T^{2} \) |
| 17 | \( 1 - 0.589T + 17T^{2} \) |
| 19 | \( 1 - 1.88T + 19T^{2} \) |
| 23 | \( 1 - 3.94T + 23T^{2} \) |
| 29 | \( 1 - 5.02T + 29T^{2} \) |
| 31 | \( 1 - 4.98T + 31T^{2} \) |
| 37 | \( 1 - 2.31T + 37T^{2} \) |
| 43 | \( 1 + 2.62T + 43T^{2} \) |
| 47 | \( 1 - 5.13T + 47T^{2} \) |
| 53 | \( 1 + 5.23T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 1.01T + 61T^{2} \) |
| 67 | \( 1 + 3.46T + 67T^{2} \) |
| 71 | \( 1 - 4.80T + 71T^{2} \) |
| 73 | \( 1 + 0.596T + 73T^{2} \) |
| 79 | \( 1 - 4.43T + 79T^{2} \) |
| 83 | \( 1 - 3.79T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−9.561706191956515968798885189662, −8.831916969561028758157803918684, −6.96657751202968977335196294391, −6.41019646798638658094285365854, −6.13695190551693786948503923196, −5.32791713298759464308925140782, −4.61105593436155717907295001650, −3.57186511262653054624283758696, −2.59914013560595730219185539056, −1.14517985020420347605453466091,
1.14517985020420347605453466091, 2.59914013560595730219185539056, 3.57186511262653054624283758696, 4.61105593436155717907295001650, 5.32791713298759464308925140782, 6.13695190551693786948503923196, 6.41019646798638658094285365854, 6.96657751202968977335196294391, 8.831916969561028758157803918684, 9.561706191956515968798885189662