| L(s) = 1 | + 3.34·2-s + 3.20·4-s + 5·5-s − 11.8·7-s − 16.0·8-s + 16.7·10-s − 13.7·11-s + 76.6·13-s − 39.5·14-s − 79.3·16-s + 21.9·17-s + 11.7·19-s + 16.0·20-s − 46.0·22-s − 184.·23-s + 25·25-s + 256.·26-s − 37.8·28-s + 29·29-s + 181.·31-s − 137.·32-s + 73.5·34-s − 59.0·35-s − 61.5·37-s + 39.2·38-s − 80.2·40-s − 465.·41-s + ⋯ |
| L(s) = 1 | + 1.18·2-s + 0.400·4-s + 0.447·5-s − 0.637·7-s − 0.709·8-s + 0.529·10-s − 0.377·11-s + 1.63·13-s − 0.754·14-s − 1.24·16-s + 0.313·17-s + 0.141·19-s + 0.179·20-s − 0.446·22-s − 1.67·23-s + 0.200·25-s + 1.93·26-s − 0.255·28-s + 0.185·29-s + 1.05·31-s − 0.757·32-s + 0.370·34-s − 0.285·35-s − 0.273·37-s + 0.167·38-s − 0.317·40-s − 1.77·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 - 3.34T + 8T^{2} \) |
| 7 | \( 1 + 11.8T + 343T^{2} \) |
| 11 | \( 1 + 13.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 76.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 21.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 11.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 184.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 181.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 61.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 465.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 410.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 307.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 371.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 287.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 421.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 899.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 37.8T + 3.57e5T^{2} \) |
| 73 | \( 1 + 859.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 30.2T + 5.71e5T^{2} \) |
| 89 | \( 1 + 231.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 513.T + 9.12e5T^{2} \) |
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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
−8.760921341723030049590548568603, −8.194251705567657574044888290487, −6.76593458546728725020806396249, −6.15606514368575990032593229664, −5.56623804264242254151580805932, −4.56165276133481271630568171951, −3.61209468487397329655486774091, −2.98501436474733770514100616856, −1.62486669995802706816280226097, 0,
1.62486669995802706816280226097, 2.98501436474733770514100616856, 3.61209468487397329655486774091, 4.56165276133481271630568171951, 5.56623804264242254151580805932, 6.15606514368575990032593229664, 6.76593458546728725020806396249, 8.194251705567657574044888290487, 8.760921341723030049590548568603
