| L(s) = 1 | − 1.41·5-s + 4·7-s + 2.82·11-s + 4.24·13-s − 2.82·19-s − 4·23-s − 2.99·25-s − 4.24·29-s − 8·31-s − 5.65·35-s − 1.41·37-s − 8·41-s + 2.82·43-s − 8·47-s + 9·49-s − 1.41·53-s − 4.00·55-s − 8.48·59-s − 4.24·61-s − 6·65-s + 2.82·67-s − 12·71-s − 2·73-s + 11.3·77-s − 14.1·83-s − 14·89-s + 16.9·91-s + ⋯ |
| L(s) = 1 | − 0.632·5-s + 1.51·7-s + 0.852·11-s + 1.17·13-s − 0.648·19-s − 0.834·23-s − 0.599·25-s − 0.787·29-s − 1.43·31-s − 0.956·35-s − 0.232·37-s − 1.24·41-s + 0.431·43-s − 1.16·47-s + 1.28·49-s − 0.194·53-s − 0.539·55-s − 1.10·59-s − 0.543·61-s − 0.744·65-s + 0.345·67-s − 1.42·71-s − 0.234·73-s + 1.28·77-s − 1.55·83-s − 1.48·89-s + 1.77·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 1.41T + 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - 2.82T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 1.41T + 53T^{2} \) |
| 59 | \( 1 + 8.48T + 59T^{2} \) |
| 61 | \( 1 + 4.24T + 61T^{2} \) |
| 67 | \( 1 - 2.82T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 16T + 97T^{2} \) |
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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
−7.53413814462915810572819498920, −6.75011339841494960825337771469, −5.94791332206254764964182323358, −5.34910991200909398195537998534, −4.33692588222372219242358122420, −4.03975531900275793108932488479, −3.23535578129751299135920327957, −1.77467576648189249407772828247, −1.54489307620531675471409139468, 0,
1.54489307620531675471409139468, 1.77467576648189249407772828247, 3.23535578129751299135920327957, 4.03975531900275793108932488479, 4.33692588222372219242358122420, 5.34910991200909398195537998534, 5.94791332206254764964182323358, 6.75011339841494960825337771469, 7.53413814462915810572819498920
