| L(s) = 1 | + 1.53·2-s + 0.369·4-s + 5-s − 7-s − 2.51·8-s + 1.53·10-s + 11-s − 0.0917·13-s − 1.53·14-s − 4.60·16-s + 5.51·17-s + 0.921·19-s + 0.369·20-s + 1.53·22-s + 5.70·23-s + 25-s − 0.141·26-s − 0.369·28-s − 1.41·29-s + 0.879·31-s − 2.06·32-s + 8.48·34-s − 35-s − 8.78·37-s + 1.41·38-s − 2.51·40-s + 1.61·41-s + ⋯ |
| L(s) = 1 | + 1.08·2-s + 0.184·4-s + 0.447·5-s − 0.377·7-s − 0.887·8-s + 0.486·10-s + 0.301·11-s − 0.0254·13-s − 0.411·14-s − 1.15·16-s + 1.33·17-s + 0.211·19-s + 0.0825·20-s + 0.328·22-s + 1.19·23-s + 0.200·25-s − 0.0276·26-s − 0.0697·28-s − 0.263·29-s + 0.157·31-s − 0.364·32-s + 1.45·34-s − 0.169·35-s − 1.44·37-s + 0.230·38-s − 0.396·40-s + 0.252·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.185975697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.185975697\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 1.53T + 2T^{2} \) |
| 13 | \( 1 + 0.0917T + 13T^{2} \) |
| 17 | \( 1 - 5.51T + 17T^{2} \) |
| 19 | \( 1 - 0.921T + 19T^{2} \) |
| 23 | \( 1 - 5.70T + 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 - 0.879T + 31T^{2} \) |
| 37 | \( 1 + 8.78T + 37T^{2} \) |
| 41 | \( 1 - 1.61T + 41T^{2} \) |
| 43 | \( 1 - 3.86T + 43T^{2} \) |
| 47 | \( 1 - 5.90T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 2.14T + 59T^{2} \) |
| 61 | \( 1 + 3.03T + 61T^{2} \) |
| 67 | \( 1 + 1.52T + 67T^{2} \) |
| 71 | \( 1 + 4.09T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 8.52T + 83T^{2} \) |
| 89 | \( 1 - 2.83T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−8.802925982292968138404453535230, −7.67148803300006672851620716905, −6.88336725865133460303460562945, −6.13485892879111480087646981161, −5.42687708521882048325304826956, −4.92967487980352627595712034733, −3.82288796543437886329939917888, −3.29189986539347761577069127263, −2.35094564977088114275337492751, −0.918374779151829555551378935217,
0.918374779151829555551378935217, 2.35094564977088114275337492751, 3.29189986539347761577069127263, 3.82288796543437886329939917888, 4.92967487980352627595712034733, 5.42687708521882048325304826956, 6.13485892879111480087646981161, 6.88336725865133460303460562945, 7.67148803300006672851620716905, 8.802925982292968138404453535230
