| L(s) = 1 | − 1.52·5-s + 7-s − 5.50·11-s + 1.15·13-s + 2.98·17-s + 19-s − 0.841·23-s − 2.68·25-s + 7.95·29-s + 6.66·31-s − 1.52·35-s + 9.49·37-s − 9.95·41-s − 1.41·43-s − 0.635·47-s + 49-s − 0.348·53-s + 8.38·55-s − 14.0·59-s − 7.01·61-s − 1.76·65-s − 3.30·67-s − 14.5·71-s − 2.26·73-s − 5.50·77-s − 6.62·79-s − 2.33·83-s + ⋯ |
| L(s) = 1 | − 0.681·5-s + 0.377·7-s − 1.66·11-s + 0.321·13-s + 0.723·17-s + 0.229·19-s − 0.175·23-s − 0.536·25-s + 1.47·29-s + 1.19·31-s − 0.257·35-s + 1.56·37-s − 1.55·41-s − 0.215·43-s − 0.0926·47-s + 0.142·49-s − 0.0478·53-s + 1.13·55-s − 1.83·59-s − 0.898·61-s − 0.218·65-s − 0.403·67-s − 1.72·71-s − 0.265·73-s − 0.627·77-s − 0.744·79-s − 0.256·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.416629328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.416629328\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 1.52T + 5T^{2} \) |
| 11 | \( 1 + 5.50T + 11T^{2} \) |
| 13 | \( 1 - 1.15T + 13T^{2} \) |
| 17 | \( 1 - 2.98T + 17T^{2} \) |
| 23 | \( 1 + 0.841T + 23T^{2} \) |
| 29 | \( 1 - 7.95T + 29T^{2} \) |
| 31 | \( 1 - 6.66T + 31T^{2} \) |
| 37 | \( 1 - 9.49T + 37T^{2} \) |
| 41 | \( 1 + 9.95T + 41T^{2} \) |
| 43 | \( 1 + 1.41T + 43T^{2} \) |
| 47 | \( 1 + 0.635T + 47T^{2} \) |
| 53 | \( 1 + 0.348T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 + 7.01T + 61T^{2} \) |
| 67 | \( 1 + 3.30T + 67T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 + 2.26T + 73T^{2} \) |
| 79 | \( 1 + 6.62T + 79T^{2} \) |
| 83 | \( 1 + 2.33T + 83T^{2} \) |
| 89 | \( 1 + 1.56T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−7.84710648899811395287876007003, −7.22418035987037389581103601057, −6.21730255434278733918236775820, −5.67889813570994895541651411063, −4.72664271595746915383840275200, −4.45361681407277393886261064847, −3.21126052220475045015685527397, −2.84766843488847437882785941233, −1.68728560422705190822543085339, −0.56770108374500450041248198734,
0.56770108374500450041248198734, 1.68728560422705190822543085339, 2.84766843488847437882785941233, 3.21126052220475045015685527397, 4.45361681407277393886261064847, 4.72664271595746915383840275200, 5.67889813570994895541651411063, 6.21730255434278733918236775820, 7.22418035987037389581103601057, 7.84710648899811395287876007003
