| L(s) = 1 | + 0.492·3-s + 0.923·5-s + 1.84·7-s − 2.75·9-s + 11-s − 6.93·13-s + 0.454·15-s − 2.04·17-s + 3.74·19-s + 0.907·21-s + 9.37·23-s − 4.14·25-s − 2.83·27-s − 9.05·29-s − 2.98·31-s + 0.492·33-s + 1.70·35-s + 4.12·37-s − 3.41·39-s − 1.47·41-s + 4.51·43-s − 2.54·45-s − 8.14·47-s − 3.60·49-s − 1.00·51-s + 53-s + 0.923·55-s + ⋯ |
| L(s) = 1 | + 0.284·3-s + 0.413·5-s + 0.696·7-s − 0.919·9-s + 0.301·11-s − 1.92·13-s + 0.117·15-s − 0.496·17-s + 0.858·19-s + 0.198·21-s + 1.95·23-s − 0.829·25-s − 0.545·27-s − 1.68·29-s − 0.535·31-s + 0.0857·33-s + 0.287·35-s + 0.678·37-s − 0.546·39-s − 0.230·41-s + 0.688·43-s − 0.379·45-s − 1.18·47-s − 0.514·49-s − 0.141·51-s + 0.137·53-s + 0.124·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 - 0.492T + 3T^{2} \) |
| 5 | \( 1 - 0.923T + 5T^{2} \) |
| 7 | \( 1 - 1.84T + 7T^{2} \) |
| 13 | \( 1 + 6.93T + 13T^{2} \) |
| 17 | \( 1 + 2.04T + 17T^{2} \) |
| 19 | \( 1 - 3.74T + 19T^{2} \) |
| 23 | \( 1 - 9.37T + 23T^{2} \) |
| 29 | \( 1 + 9.05T + 29T^{2} \) |
| 31 | \( 1 + 2.98T + 31T^{2} \) |
| 37 | \( 1 - 4.12T + 37T^{2} \) |
| 41 | \( 1 + 1.47T + 41T^{2} \) |
| 43 | \( 1 - 4.51T + 43T^{2} \) |
| 47 | \( 1 + 8.14T + 47T^{2} \) |
| 59 | \( 1 + 2.71T + 59T^{2} \) |
| 61 | \( 1 + 4.46T + 61T^{2} \) |
| 67 | \( 1 - 1.77T + 67T^{2} \) |
| 71 | \( 1 - 4.40T + 71T^{2} \) |
| 73 | \( 1 - 0.530T + 73T^{2} \) |
| 79 | \( 1 + 7.81T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 - 0.0829T + 89T^{2} \) |
| 97 | \( 1 + 8.81T + 97T^{2} \) |
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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
−7.78885642761804901344460792613, −7.41386163376154692404448163973, −6.57931878320141824736618831373, −5.43859449799834074726775235653, −5.21713762736308241979851240842, −4.26129648004100803950117619163, −3.13479263907067278894530513939, −2.45499795690449180678170496626, −1.55893714002353409632701844106, 0,
1.55893714002353409632701844106, 2.45499795690449180678170496626, 3.13479263907067278894530513939, 4.26129648004100803950117619163, 5.21713762736308241979851240842, 5.43859449799834074726775235653, 6.57931878320141824736618831373, 7.41386163376154692404448163973, 7.78885642761804901344460792613
