| L(s) = 1 | + 7-s − 4·11-s + 13-s − 5·17-s − 2·19-s + 3·23-s − 5·25-s + 29-s + 9·31-s + 6·37-s + 2·41-s − 43-s − 2·47-s + 49-s − 9·53-s − 3·59-s − 2·61-s + 11·67-s + 5·71-s − 2·73-s − 4·77-s + 14·79-s + 12·83-s + 15·89-s + 91-s + 12·97-s + 6·101-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.20·11-s + 0.277·13-s − 1.21·17-s − 0.458·19-s + 0.625·23-s − 25-s + 0.185·29-s + 1.61·31-s + 0.986·37-s + 0.312·41-s − 0.152·43-s − 0.291·47-s + 1/7·49-s − 1.23·53-s − 0.390·59-s − 0.256·61-s + 1.34·67-s + 0.593·71-s − 0.234·73-s − 0.455·77-s + 1.57·79-s + 1.31·83-s + 1.58·89-s + 0.104·91-s + 1.21·97-s + 0.597·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.623450144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.623450144\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.014024272160445569125341691697, −7.56925164639793955738726327455, −6.49158587759524874508499456433, −6.11389038297671043081241088153, −4.97045257499060040224701679220, −4.66451666689748678252232989883, −3.64719695497034750400463779919, −2.64297116556229739712532891812, −2.00812423463094579156301367237, −0.65023209738077206256465214358,
0.65023209738077206256465214358, 2.00812423463094579156301367237, 2.64297116556229739712532891812, 3.64719695497034750400463779919, 4.66451666689748678252232989883, 4.97045257499060040224701679220, 6.11389038297671043081241088153, 6.49158587759524874508499456433, 7.56925164639793955738726327455, 8.014024272160445569125341691697
