| L(s) = 1 | − 2.37·5-s − 4.37·7-s + 11-s − 0.372·13-s + 5.37·17-s − 0.627·19-s − 0.372·23-s + 0.627·25-s + 4.37·29-s − 6.37·31-s + 10.3·35-s + 8.74·37-s + 11.7·41-s + 1.74·43-s + 4.37·47-s + 12.1·49-s − 0.744·53-s − 2.37·55-s + 7·59-s + 2.37·61-s + 0.883·65-s + 3.74·67-s − 4·71-s − 12.1·73-s − 4.37·77-s + 6.37·79-s + 9.62·83-s + ⋯ |
| L(s) = 1 | − 1.06·5-s − 1.65·7-s + 0.301·11-s − 0.103·13-s + 1.30·17-s − 0.144·19-s − 0.0776·23-s + 0.125·25-s + 0.811·29-s − 1.14·31-s + 1.75·35-s + 1.43·37-s + 1.83·41-s + 0.266·43-s + 0.637·47-s + 1.73·49-s − 0.102·53-s − 0.319·55-s + 0.911·59-s + 0.303·61-s + 0.109·65-s + 0.457·67-s − 0.474·71-s − 1.41·73-s − 0.498·77-s + 0.716·79-s + 1.05·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9687228067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9687228067\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2.37T + 5T^{2} \) |
| 7 | \( 1 + 4.37T + 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + 0.372T + 13T^{2} \) |
| 17 | \( 1 - 5.37T + 17T^{2} \) |
| 19 | \( 1 + 0.627T + 19T^{2} \) |
| 23 | \( 1 + 0.372T + 23T^{2} \) |
| 29 | \( 1 - 4.37T + 29T^{2} \) |
| 31 | \( 1 + 6.37T + 31T^{2} \) |
| 37 | \( 1 - 8.74T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 1.74T + 43T^{2} \) |
| 47 | \( 1 - 4.37T + 47T^{2} \) |
| 53 | \( 1 + 0.744T + 53T^{2} \) |
| 59 | \( 1 - 7T + 59T^{2} \) |
| 61 | \( 1 - 2.37T + 61T^{2} \) |
| 67 | \( 1 - 3.74T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 6.37T + 79T^{2} \) |
| 83 | \( 1 - 9.62T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 1.74T + 97T^{2} \) |
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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
−9.631776519077411429459061501138, −8.965702564228445759132617885597, −7.86370042178954888050792310545, −7.30966098815369833582236284677, −6.36121670194464273730365479425, −5.62736689584681995744058291532, −4.22633217333827336929026314240, −3.58184897069210958814638415701, −2.70261549260161757630891647851, −0.70592632603964084143015223490,
0.70592632603964084143015223490, 2.70261549260161757630891647851, 3.58184897069210958814638415701, 4.22633217333827336929026314240, 5.62736689584681995744058291532, 6.36121670194464273730365479425, 7.30966098815369833582236284677, 7.86370042178954888050792310545, 8.965702564228445759132617885597, 9.631776519077411429459061501138
