| L(s) = 1 | − 7-s − 3·9-s − 4·11-s + 8·19-s − 23-s + 6·29-s − 4·31-s + 8·37-s + 6·41-s + 8·47-s + 49-s + 4·59-s − 10·61-s + 3·63-s + 4·73-s + 4·77-s + 4·79-s + 9·81-s − 8·83-s + 6·89-s − 8·97-s + 12·99-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s − 1.20·11-s + 1.83·19-s − 0.208·23-s + 1.11·29-s − 0.718·31-s + 1.31·37-s + 0.937·41-s + 1.16·47-s + 1/7·49-s + 0.520·59-s − 1.28·61-s + 0.377·63-s + 0.468·73-s + 0.455·77-s + 0.450·79-s + 81-s − 0.878·83-s + 0.635·89-s − 0.812·97-s + 1.20·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.17434193077630, −12.47185019517492, −12.18717267330624, −11.70902191539589, −11.13889696895556, −10.81939484064990, −10.33109638116141, −9.714740044597722, −9.426708058871879, −8.918361770451497, −8.312053039988429, −7.878883992128557, −7.509909576538752, −7.025906965529549, −6.292077546771142, −5.844248670144602, −5.461156377982961, −5.017652354630597, −4.410555441926815, −3.703933455361868, −3.171308285208257, −2.571659069063165, −2.478382053730821, −1.323355863536517, −0.7333139029850949, 0,
0.7333139029850949, 1.323355863536517, 2.478382053730821, 2.571659069063165, 3.171308285208257, 3.703933455361868, 4.410555441926815, 5.017652354630597, 5.461156377982961, 5.844248670144602, 6.292077546771142, 7.025906965529549, 7.509909576538752, 7.878883992128557, 8.312053039988429, 8.918361770451497, 9.426708058871879, 9.714740044597722, 10.33109638116141, 10.81939484064990, 11.13889696895556, 11.70902191539589, 12.18717267330624, 12.47185019517492, 13.17434193077630
