| L(s) = 1 | + 5-s − 7-s − 3·9-s + 11-s − 4·19-s + 9·23-s − 4·25-s + 3·29-s − 8·31-s − 35-s + 2·37-s + 5·41-s − 3·43-s − 3·45-s + 2·47-s − 6·49-s − 2·53-s + 55-s + 7·59-s + 7·61-s + 3·63-s + 7·67-s − 10·71-s + 7·73-s − 77-s + 8·79-s + 9·81-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 9-s + 0.301·11-s − 0.917·19-s + 1.87·23-s − 4/5·25-s + 0.557·29-s − 1.43·31-s − 0.169·35-s + 0.328·37-s + 0.780·41-s − 0.457·43-s − 0.447·45-s + 0.291·47-s − 6/7·49-s − 0.274·53-s + 0.134·55-s + 0.911·59-s + 0.896·61-s + 0.377·63-s + 0.855·67-s − 1.18·71-s + 0.819·73-s − 0.113·77-s + 0.900·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.28058420586894, −14.78231803234547, −14.46032364307019, −13.82219155275147, −13.27787666378509, −12.76943460758515, −12.39545571919436, −11.41310626923798, −11.28704108103499, −10.64546163303745, −9.990572347152229, −9.351274200909387, −8.994208864897701, −8.415498134150059, −7.812125340686489, −6.970198389218643, −6.591386051545948, −5.893089698624511, −5.425280406952841, −4.780741675417987, −3.939401529870317, −3.307038197333520, −2.619491194378952, −1.987541064864655, −0.9876501945790397, 0,
0.9876501945790397, 1.987541064864655, 2.619491194378952, 3.307038197333520, 3.939401529870317, 4.780741675417987, 5.425280406952841, 5.893089698624511, 6.591386051545948, 6.970198389218643, 7.812125340686489, 8.415498134150059, 8.994208864897701, 9.351274200909387, 9.990572347152229, 10.64546163303745, 11.28704108103499, 11.41310626923798, 12.39545571919436, 12.76943460758515, 13.27787666378509, 13.82219155275147, 14.46032364307019, 14.78231803234547, 15.28058420586894
