| L(s) = 1 | − 1.19·3-s + 5-s + 1.83·7-s − 1.57·9-s − 1.25·13-s − 1.19·15-s + 6.44·17-s − 1.25·19-s − 2.19·21-s + 4.35·23-s + 25-s + 5.46·27-s − 0.726·29-s + 11.0·31-s + 1.83·35-s − 11.5·37-s + 1.49·39-s + 7.92·41-s − 0.606·43-s − 1.57·45-s + 3.98·47-s − 3.62·49-s − 7.69·51-s + 6.83·53-s + 1.49·57-s + 1.41·59-s − 9.75·61-s + ⋯ |
| L(s) = 1 | − 0.689·3-s + 0.447·5-s + 0.694·7-s − 0.525·9-s − 0.348·13-s − 0.308·15-s + 1.56·17-s − 0.287·19-s − 0.478·21-s + 0.908·23-s + 0.200·25-s + 1.05·27-s − 0.134·29-s + 1.98·31-s + 0.310·35-s − 1.89·37-s + 0.239·39-s + 1.23·41-s − 0.0925·43-s − 0.234·45-s + 0.580·47-s − 0.517·49-s − 1.07·51-s + 0.938·53-s + 0.197·57-s + 0.183·59-s − 1.24·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.898329427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.898329427\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 1.19T + 3T^{2} \) |
| 7 | \( 1 - 1.83T + 7T^{2} \) |
| 13 | \( 1 + 1.25T + 13T^{2} \) |
| 17 | \( 1 - 6.44T + 17T^{2} \) |
| 19 | \( 1 + 1.25T + 19T^{2} \) |
| 23 | \( 1 - 4.35T + 23T^{2} \) |
| 29 | \( 1 + 0.726T + 29T^{2} \) |
| 31 | \( 1 - 11.0T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 7.92T + 41T^{2} \) |
| 43 | \( 1 + 0.606T + 43T^{2} \) |
| 47 | \( 1 - 3.98T + 47T^{2} \) |
| 53 | \( 1 - 6.83T + 53T^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 + 9.75T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 - 6.79T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 6.51T + 79T^{2} \) |
| 83 | \( 1 + 0.644T + 83T^{2} \) |
| 89 | \( 1 - 9.39T + 89T^{2} \) |
| 97 | \( 1 + 4.66T + 97T^{2} \) |
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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
−7.62653144850063123046371048284, −6.98244811611616955712912791263, −6.13032095973299476188867201271, −5.65457612942879947279638454250, −5.01259683942816346498880063252, −4.47959263966494518618954879360, −3.28869609324966991752454080084, −2.65748338701938155029303816009, −1.55527324735376718125687627928, −0.71804423399374604519319968867,
0.71804423399374604519319968867, 1.55527324735376718125687627928, 2.65748338701938155029303816009, 3.28869609324966991752454080084, 4.47959263966494518618954879360, 5.01259683942816346498880063252, 5.65457612942879947279638454250, 6.13032095973299476188867201271, 6.98244811611616955712912791263, 7.62653144850063123046371048284
