| L(s) = 1 | − 7-s − 4·11-s − 2·17-s + 2·19-s + 6·23-s − 4·29-s + 2·31-s − 10·37-s − 10·41-s − 4·43-s + 4·47-s + 49-s + 2·53-s + 12·59-s + 10·61-s + 12·67-s − 4·73-s + 4·77-s − 16·79-s + 4·83-s + 6·89-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.20·11-s − 0.485·17-s + 0.458·19-s + 1.25·23-s − 0.742·29-s + 0.359·31-s − 1.64·37-s − 1.56·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.274·53-s + 1.56·59-s + 1.28·61-s + 1.46·67-s − 0.468·73-s + 0.455·77-s − 1.80·79-s + 0.439·83-s + 0.635·89-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−13.87049901226861, −13.28832352898302, −13.20018985338884, −12.64140825219022, −12.00611202913546, −11.56427797371292, −11.01946000920764, −10.53362299533380, −10.03209068546207, −9.710660800860554, −8.874025029510257, −8.609101099938209, −8.075045207349970, −7.335587704748971, −6.969169121979892, −6.581142997527837, −5.684660724764296, −5.219200151861411, −4.997617625502962, −4.074256869411562, −3.497613641487606, −2.961939347568909, −2.349130779213240, −1.696322568055473, −0.7590803564613323, 0,
0.7590803564613323, 1.696322568055473, 2.349130779213240, 2.961939347568909, 3.497613641487606, 4.074256869411562, 4.997617625502962, 5.219200151861411, 5.684660724764296, 6.581142997527837, 6.969169121979892, 7.335587704748971, 8.075045207349970, 8.609101099938209, 8.874025029510257, 9.710660800860554, 10.03209068546207, 10.53362299533380, 11.01946000920764, 11.56427797371292, 12.00611202913546, 12.64140825219022, 13.20018985338884, 13.28832352898302, 13.87049901226861
