| L(s) = 1 | − 2.49·2-s − 0.713·3-s + 4.20·4-s + 2.20·5-s + 1.77·6-s − 5.49·8-s − 2.49·9-s − 5.49·10-s − 11-s − 2.99·12-s − 3.28·13-s − 1.57·15-s + 5.26·16-s + 1.49·17-s + 6.20·18-s − 6.91·19-s + 9.26·20-s + 2.49·22-s + 6.49·23-s + 3.91·24-s − 0.140·25-s + 8.18·26-s + 3.91·27-s − 1.64·29-s + 3.91·30-s − 2.35·31-s − 2.14·32-s + ⋯ |
| L(s) = 1 | − 1.76·2-s − 0.411·3-s + 2.10·4-s + 0.985·5-s + 0.725·6-s − 1.94·8-s − 0.830·9-s − 1.73·10-s − 0.301·11-s − 0.866·12-s − 0.911·13-s − 0.406·15-s + 1.31·16-s + 0.361·17-s + 1.46·18-s − 1.58·19-s + 2.07·20-s + 0.531·22-s + 1.35·23-s + 0.799·24-s − 0.0281·25-s + 1.60·26-s + 0.754·27-s − 0.306·29-s + 0.715·30-s − 0.422·31-s − 0.378·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.49T + 2T^{2} \) |
| 3 | \( 1 + 0.713T + 3T^{2} \) |
| 5 | \( 1 - 2.20T + 5T^{2} \) |
| 13 | \( 1 + 3.28T + 13T^{2} \) |
| 17 | \( 1 - 1.49T + 17T^{2} \) |
| 19 | \( 1 + 6.91T + 19T^{2} \) |
| 23 | \( 1 - 6.49T + 23T^{2} \) |
| 29 | \( 1 + 1.64T + 29T^{2} \) |
| 31 | \( 1 + 2.35T + 31T^{2} \) |
| 37 | \( 1 + 5.55T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 5.26T + 43T^{2} \) |
| 47 | \( 1 - 1.49T + 47T^{2} \) |
| 53 | \( 1 - 0.304T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 + 4.57T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 8.56T + 73T^{2} \) |
| 79 | \( 1 - 4.63T + 79T^{2} \) |
| 83 | \( 1 + 1.93T + 83T^{2} \) |
| 89 | \( 1 - 3.20T + 89T^{2} \) |
| 97 | \( 1 - 1.85T + 97T^{2} \) |
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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
−10.45223431127768264278935458379, −9.370719277056865651776781881010, −8.859074389249418760599333235781, −7.88847781648112056309951457175, −6.88639247848090395940145970180, −6.07931724988479289282338665983, −5.06192667716581458177367709659, −2.82082233898832548055628883683, −1.78477482011343864942782577083, 0,
1.78477482011343864942782577083, 2.82082233898832548055628883683, 5.06192667716581458177367709659, 6.07931724988479289282338665983, 6.88639247848090395940145970180, 7.88847781648112056309951457175, 8.859074389249418760599333235781, 9.370719277056865651776781881010, 10.45223431127768264278935458379
