| L(s) = 1 | − 2.82·5-s + 7-s − 3·9-s + 2.82·11-s + 2.82·13-s − 2·17-s + 5.65·19-s − 8·23-s + 3.00·25-s + 5.65·29-s − 8·31-s − 2.82·35-s − 5.65·37-s + 6·41-s + 2.82·43-s + 8.48·45-s − 8·47-s + 49-s − 11.3·53-s − 8.00·55-s − 11.3·59-s − 2.82·61-s − 3·63-s − 8.00·65-s − 8.48·67-s − 8·71-s − 6·73-s + ⋯ |
| L(s) = 1 | − 1.26·5-s + 0.377·7-s − 9-s + 0.852·11-s + 0.784·13-s − 0.485·17-s + 1.29·19-s − 1.66·23-s + 0.600·25-s + 1.05·29-s − 1.43·31-s − 0.478·35-s − 0.929·37-s + 0.937·41-s + 0.431·43-s + 1.26·45-s − 1.16·47-s + 0.142·49-s − 1.55·53-s − 1.07·55-s − 1.47·59-s − 0.362·61-s − 0.377·63-s − 0.992·65-s − 1.03·67-s − 0.949·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 2.82T + 5T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 - 5.65T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 5.65T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 2.82T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 2.82T + 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−8.799236922178969684769191168121, −8.045245190638781218099304120346, −7.54412029242978308809836179681, −6.45630596817959378454372774453, −5.71630079190482484194003814921, −4.60840781381815747813842834000, −3.80060917499115035392136701315, −3.08307298683603365154117235205, −1.54194435066604884302410940401, 0,
1.54194435066604884302410940401, 3.08307298683603365154117235205, 3.80060917499115035392136701315, 4.60840781381815747813842834000, 5.71630079190482484194003814921, 6.45630596817959378454372774453, 7.54412029242978308809836179681, 8.045245190638781218099304120346, 8.799236922178969684769191168121
