| L(s) = 1 | − 1.27·3-s − 19.8·5-s + 7·7-s − 25.3·9-s + 11·11-s − 35.4·13-s + 25.2·15-s + 65.1·17-s + 83.0·19-s − 8.92·21-s + 70.7·23-s + 268.·25-s + 66.7·27-s + 243.·29-s + 69.4·31-s − 14.0·33-s − 138.·35-s − 131.·37-s + 45.1·39-s − 482.·41-s − 498.·43-s + 503.·45-s + 504.·47-s + 49·49-s − 83.0·51-s − 573.·53-s − 218.·55-s + ⋯ |
| L(s) = 1 | − 0.245·3-s − 1.77·5-s + 0.377·7-s − 0.939·9-s + 0.301·11-s − 0.756·13-s + 0.435·15-s + 0.929·17-s + 1.00·19-s − 0.0927·21-s + 0.641·23-s + 2.14·25-s + 0.475·27-s + 1.55·29-s + 0.402·31-s − 0.0739·33-s − 0.670·35-s − 0.585·37-s + 0.185·39-s − 1.83·41-s − 1.76·43-s + 1.66·45-s + 1.56·47-s + 0.142·49-s − 0.228·51-s − 1.48·53-s − 0.534·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 + 1.27T + 27T^{2} \) |
| 5 | \( 1 + 19.8T + 125T^{2} \) |
| 13 | \( 1 + 35.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 65.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 83.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 70.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 243.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 69.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 131.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 482.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 498.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 504.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 573.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 262.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 884.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 396.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 414.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 205.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 232.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 584.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 629.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 320.T + 9.12e5T^{2} \) |
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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
−8.596240282729508514910398704147, −8.194614939414958622820808412601, −7.36086694228590903178907474779, −6.64834927222021733116547255860, −5.26577134860883502587812435887, −4.74939898267623058648779762062, −3.54453773665103391522966030240, −2.91553962195133858767669820248, −1.06027715211277279575475315198, 0,
1.06027715211277279575475315198, 2.91553962195133858767669820248, 3.54453773665103391522966030240, 4.74939898267623058648779762062, 5.26577134860883502587812435887, 6.64834927222021733116547255860, 7.36086694228590903178907474779, 8.194614939414958622820808412601, 8.596240282729508514910398704147
