| L(s) = 1 | − 1.73·2-s + 0.732·3-s + 0.999·4-s + 1.73·5-s − 1.26·6-s + 1.73·8-s − 2.46·9-s − 2.99·10-s + 4.73·11-s + 0.732·12-s + 1.26·15-s − 5·16-s + 4.26·17-s + 4.26·18-s − 2·19-s + 1.73·20-s − 8.19·22-s + 1.26·23-s + 1.26·24-s − 2.00·25-s − 4·27-s − 3·29-s − 2.19·30-s + 6.19·31-s + 5.19·32-s + 3.46·33-s − 7.39·34-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.422·3-s + 0.499·4-s + 0.774·5-s − 0.517·6-s + 0.612·8-s − 0.821·9-s − 0.948·10-s + 1.42·11-s + 0.211·12-s + 0.327·15-s − 1.25·16-s + 1.03·17-s + 1.00·18-s − 0.458·19-s + 0.387·20-s − 1.74·22-s + 0.264·23-s + 0.258·24-s − 0.400·25-s − 0.769·27-s − 0.557·29-s − 0.400·30-s + 1.11·31-s + 0.918·32-s + 0.603·33-s − 1.26·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.419975230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.419975230\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 3 | \( 1 - 0.732T + 3T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 17 | \( 1 - 4.26T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 1.26T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 6.19T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + 5.19T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 0.928T + 47T^{2} \) |
| 53 | \( 1 - 3.92T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 - 4.19T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 7.19T + 73T^{2} \) |
| 79 | \( 1 - 5.80T + 79T^{2} \) |
| 83 | \( 1 + 8.19T + 83T^{2} \) |
| 89 | \( 1 - 0.928T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−7.990175628779371169469653973427, −7.32783132840581600025200395747, −6.54929689587369269511445986964, −5.91752987531568796858966322458, −5.15001605685372899395956367285, −4.12709760582645491900121620113, −3.38470388426645096976732282369, −2.32252478225906546509591562030, −1.62853272618901808335164087384, −0.72381723488836291932113148643,
0.72381723488836291932113148643, 1.62853272618901808335164087384, 2.32252478225906546509591562030, 3.38470388426645096976732282369, 4.12709760582645491900121620113, 5.15001605685372899395956367285, 5.91752987531568796858966322458, 6.54929689587369269511445986964, 7.32783132840581600025200395747, 7.990175628779371169469653973427
