| L(s) = 1 | + 2.82·3-s + 1.41·5-s − 7-s + 5.00·9-s − 5.65·11-s − 1.41·13-s + 4.00·15-s − 4·17-s + 2.82·19-s − 2.82·21-s − 4·23-s − 2.99·25-s + 5.65·27-s − 9.89·29-s − 4·31-s − 16.0·33-s − 1.41·35-s + 7.07·37-s − 4.00·39-s + 12·41-s + 7.07·45-s − 12·47-s + 49-s − 11.3·51-s − 1.41·53-s − 8.00·55-s + 8.00·57-s + ⋯ |
| L(s) = 1 | + 1.63·3-s + 0.632·5-s − 0.377·7-s + 1.66·9-s − 1.70·11-s − 0.392·13-s + 1.03·15-s − 0.970·17-s + 0.648·19-s − 0.617·21-s − 0.834·23-s − 0.599·25-s + 1.08·27-s − 1.83·29-s − 0.718·31-s − 2.78·33-s − 0.239·35-s + 1.16·37-s − 0.640·39-s + 1.87·41-s + 1.05·45-s − 1.75·47-s + 0.142·49-s − 1.58·51-s − 0.194·53-s − 1.07·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2.82T + 3T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 9.89T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 7.07T + 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 + 1.41T + 53T^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 + 4.24T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
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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
−7.70144368300105258566912689531, −7.25405312436087011642522665178, −6.12705833023410270433720664446, −5.51684503395166427126716976185, −4.58440531085820872092377460733, −3.77267439763344708111097560315, −2.94788297923571259838111865797, −2.35061266006458097771966700092, −1.80487200704830724779147036705, 0,
1.80487200704830724779147036705, 2.35061266006458097771966700092, 2.94788297923571259838111865797, 3.77267439763344708111097560315, 4.58440531085820872092377460733, 5.51684503395166427126716976185, 6.12705833023410270433720664446, 7.25405312436087011642522665178, 7.70144368300105258566912689531
