L(s) = 1 | + 5-s + 7-s − 2·11-s + 6·17-s − 4·19-s + 23-s + 25-s − 2·29-s − 2·31-s + 35-s + 2·41-s − 4·43-s − 4·47-s + 49-s − 2·55-s + 6·59-s + 6·61-s − 8·67-s + 12·71-s − 16·73-s − 2·77-s + 14·79-s − 4·83-s + 6·85-s + 6·89-s − 4·95-s + 18·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s − 0.603·11-s + 1.45·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s − 0.371·29-s − 0.359·31-s + 0.169·35-s + 0.312·41-s − 0.609·43-s − 0.583·47-s + 1/7·49-s − 0.269·55-s + 0.781·59-s + 0.768·61-s − 0.977·67-s + 1.42·71-s − 1.87·73-s − 0.227·77-s + 1.57·79-s − 0.439·83-s + 0.650·85-s + 0.635·89-s − 0.410·95-s + 1.82·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.84456638561633, −13.29025802284135, −12.88764250065695, −12.53689336257536, −11.78065980191058, −11.56716721714362, −10.76855063744355, −10.44837284059158, −10.04768978967778, −9.446967221158470, −8.953532947369846, −8.405065759156169, −7.806068312073224, −7.595985603206286, −6.771082207210420, −6.360621407394934, −5.653234845143625, −5.288565729970645, −4.829288103531927, −4.051458607597956, −3.527536909384185, −2.846213564275542, −2.250062144290115, −1.619208361706768, −0.9408287613334088, 0,
0.9408287613334088, 1.619208361706768, 2.250062144290115, 2.846213564275542, 3.527536909384185, 4.051458607597956, 4.829288103531927, 5.288565729970645, 5.653234845143625, 6.360621407394934, 6.771082207210420, 7.595985603206286, 7.806068312073224, 8.405065759156169, 8.953532947369846, 9.446967221158470, 10.04768978967778, 10.44837284059158, 10.76855063744355, 11.56716721714362, 11.78065980191058, 12.53689336257536, 12.88764250065695, 13.29025802284135, 13.84456638561633