| L(s) = 1 | + 2.49·3-s + 1.95·5-s − 4.15·7-s + 3.22·9-s + 5.07·11-s − 3.39·13-s + 4.88·15-s − 17-s − 7.96·19-s − 10.3·21-s − 1.52·23-s − 1.16·25-s + 0.559·27-s + 2.94·29-s − 9.57·31-s + 12.6·33-s − 8.14·35-s − 3.43·37-s − 8.47·39-s + 1.52·41-s − 0.867·43-s + 6.31·45-s − 1.96·47-s + 10.2·49-s − 2.49·51-s + 3.60·53-s + 9.93·55-s + ⋯ |
| L(s) = 1 | + 1.44·3-s + 0.875·5-s − 1.57·7-s + 1.07·9-s + 1.52·11-s − 0.941·13-s + 1.26·15-s − 0.242·17-s − 1.82·19-s − 2.26·21-s − 0.317·23-s − 0.232·25-s + 0.107·27-s + 0.546·29-s − 1.71·31-s + 2.20·33-s − 1.37·35-s − 0.564·37-s − 1.35·39-s + 0.237·41-s − 0.132·43-s + 0.941·45-s − 0.286·47-s + 1.46·49-s − 0.349·51-s + 0.495·53-s + 1.34·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 2.49T + 3T^{2} \) |
| 5 | \( 1 - 1.95T + 5T^{2} \) |
| 7 | \( 1 + 4.15T + 7T^{2} \) |
| 11 | \( 1 - 5.07T + 11T^{2} \) |
| 13 | \( 1 + 3.39T + 13T^{2} \) |
| 19 | \( 1 + 7.96T + 19T^{2} \) |
| 23 | \( 1 + 1.52T + 23T^{2} \) |
| 29 | \( 1 - 2.94T + 29T^{2} \) |
| 31 | \( 1 + 9.57T + 31T^{2} \) |
| 37 | \( 1 + 3.43T + 37T^{2} \) |
| 41 | \( 1 - 1.52T + 41T^{2} \) |
| 43 | \( 1 + 0.867T + 43T^{2} \) |
| 47 | \( 1 + 1.96T + 47T^{2} \) |
| 53 | \( 1 - 3.60T + 53T^{2} \) |
| 61 | \( 1 + 2.74T + 61T^{2} \) |
| 67 | \( 1 - 16.0T + 67T^{2} \) |
| 71 | \( 1 - 1.96T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 + 9.09T + 79T^{2} \) |
| 83 | \( 1 + 17.0T + 83T^{2} \) |
| 89 | \( 1 - 2.26T + 89T^{2} \) |
| 97 | \( 1 - 4.97T + 97T^{2} \) |
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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
−7.37051925272639338911449418991, −6.80121170536252211512578290752, −6.30205657043062297788042467007, −5.57866774373417194761134862544, −4.26442542166365785529802361631, −3.82514766585274922277790308045, −3.02636157373010186430816521907, −2.27779372600050600744403829115, −1.70316375004063620955654181229, 0,
1.70316375004063620955654181229, 2.27779372600050600744403829115, 3.02636157373010186430816521907, 3.82514766585274922277790308045, 4.26442542166365785529802361631, 5.57866774373417194761134862544, 6.30205657043062297788042467007, 6.80121170536252211512578290752, 7.37051925272639338911449418991
