L(s) = 1 | + 4·7-s − 4·11-s − 4·13-s + 2·17-s − 4·19-s − 8·23-s − 5·25-s − 8·29-s + 4·31-s + 4·37-s − 6·41-s + 4·43-s − 8·47-s + 9·49-s − 8·53-s + 12·59-s − 12·61-s + 12·67-s + 8·71-s − 6·73-s − 16·77-s + 4·79-s + 4·83-s + 6·89-s − 16·91-s − 2·97-s − 8·101-s + ⋯ |
L(s) = 1 | + 1.51·7-s − 1.20·11-s − 1.10·13-s + 0.485·17-s − 0.917·19-s − 1.66·23-s − 25-s − 1.48·29-s + 0.718·31-s + 0.657·37-s − 0.937·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s − 1.09·53-s + 1.56·59-s − 1.53·61-s + 1.46·67-s + 0.949·71-s − 0.702·73-s − 1.82·77-s + 0.450·79-s + 0.439·83-s + 0.635·89-s − 1.67·91-s − 0.203·97-s − 0.796·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.177321188758195754539998419996, −8.047407760393037637469456944555, −7.36659210685234271732440738739, −6.14794233023491327465599035306, −5.31767247249958159060229217606, −4.73277311617358885865114786873, −3.83954596489946517246485011666, −2.42199783197145401637689570724, −1.81521099078756431407618829430, 0,
1.81521099078756431407618829430, 2.42199783197145401637689570724, 3.83954596489946517246485011666, 4.73277311617358885865114786873, 5.31767247249958159060229217606, 6.14794233023491327465599035306, 7.36659210685234271732440738739, 8.047407760393037637469456944555, 8.177321188758195754539998419996