L(s) = 1 | + 0.732·3-s − 2.73·5-s + 7-s − 2.46·9-s + 1.46·11-s − 2.73·13-s − 2·15-s + 7.46·17-s − 6.19·19-s + 0.732·21-s − 8.92·23-s + 2.46·25-s − 4·27-s − 3.46·29-s − 2.53·31-s + 1.07·33-s − 2.73·35-s − 4.53·37-s − 2·39-s − 3.46·41-s − 2.53·43-s + 6.73·45-s + 1.46·47-s + 49-s + 5.46·51-s − 6·53-s − 4·55-s + ⋯ |
L(s) = 1 | + 0.422·3-s − 1.22·5-s + 0.377·7-s − 0.821·9-s + 0.441·11-s − 0.757·13-s − 0.516·15-s + 1.81·17-s − 1.42·19-s + 0.159·21-s − 1.86·23-s + 0.492·25-s − 0.769·27-s − 0.643·29-s − 0.455·31-s + 0.186·33-s − 0.461·35-s − 0.745·37-s − 0.320·39-s − 0.541·41-s − 0.386·43-s + 1.00·45-s + 0.213·47-s + 0.142·49-s + 0.765·51-s − 0.824·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 5 | \( 1 + 2.73T + 5T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 17 | \( 1 - 7.46T + 17T^{2} \) |
| 19 | \( 1 + 6.19T + 19T^{2} \) |
| 23 | \( 1 + 8.92T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 2.53T + 31T^{2} \) |
| 37 | \( 1 + 4.53T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 + 2.53T + 43T^{2} \) |
| 47 | \( 1 - 1.46T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 6.19T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 8.92T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 7.26T + 83T^{2} \) |
| 89 | \( 1 - 4.92T + 89T^{2} \) |
| 97 | \( 1 - 0.535T + 97T^{2} \) |
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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
−9.669178550281557358504751419493, −8.623966591954662412499846625440, −7.989047184897726615632956030175, −7.50275708361587723303325108619, −6.23500742017146899702803340536, −5.22495632374253093676830123624, −4.04389016047341696325705537080, −3.39408808875610188540011035469, −2.01625353173241807079879469062, 0,
2.01625353173241807079879469062, 3.39408808875610188540011035469, 4.04389016047341696325705537080, 5.22495632374253093676830123624, 6.23500742017146899702803340536, 7.50275708361587723303325108619, 7.989047184897726615632956030175, 8.623966591954662412499846625440, 9.669178550281557358504751419493