| L(s) = 1 | − 0.414·2-s − 1.82·4-s + 5-s + 1.58·8-s − 0.414·10-s − 2.41·11-s − 6.24·13-s + 3·16-s − 4·17-s + 19-s − 1.82·20-s + 0.999·22-s + 8.41·23-s − 4·25-s + 2.58·26-s + 9.41·29-s − 2.24·31-s − 4.41·32-s + 1.65·34-s − 5.07·37-s − 0.414·38-s + 1.58·40-s − 8.82·41-s − 6.07·43-s + 4.41·44-s − 3.48·46-s − 1.58·47-s + ⋯ |
| L(s) = 1 | − 0.292·2-s − 0.914·4-s + 0.447·5-s + 0.560·8-s − 0.130·10-s − 0.727·11-s − 1.73·13-s + 0.750·16-s − 0.970·17-s + 0.229·19-s − 0.408·20-s + 0.213·22-s + 1.75·23-s − 0.800·25-s + 0.507·26-s + 1.74·29-s − 0.402·31-s − 0.780·32-s + 0.284·34-s − 0.833·37-s − 0.0671·38-s + 0.250·40-s − 1.37·41-s − 0.925·43-s + 0.665·44-s − 0.513·46-s − 0.231·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7897749449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7897749449\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 + 2.41T + 11T^{2} \) |
| 13 | \( 1 + 6.24T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 23 | \( 1 - 8.41T + 23T^{2} \) |
| 29 | \( 1 - 9.41T + 29T^{2} \) |
| 31 | \( 1 + 2.24T + 31T^{2} \) |
| 37 | \( 1 + 5.07T + 37T^{2} \) |
| 41 | \( 1 + 8.82T + 41T^{2} \) |
| 43 | \( 1 + 6.07T + 43T^{2} \) |
| 47 | \( 1 + 1.58T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 - 1.17T + 59T^{2} \) |
| 61 | \( 1 + 6.17T + 61T^{2} \) |
| 67 | \( 1 - 6.82T + 67T^{2} \) |
| 71 | \( 1 - 5.75T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 1.41T + 79T^{2} \) |
| 83 | \( 1 - 0.757T + 83T^{2} \) |
| 89 | \( 1 + 8.58T + 89T^{2} \) |
| 97 | \( 1 - 8.82T + 97T^{2} \) |
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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
−7.955847194745948757466401449731, −7.07849168213727086924731594792, −6.63984849171206317592199153186, −5.41256067580243381231398531207, −4.98711705259485142364187464826, −4.57856578128829407279366795195, −3.40532119917113448227826210075, −2.62548876893381550671462056413, −1.72563003880105871263718893999, −0.45059304586768951778828774255,
0.45059304586768951778828774255, 1.72563003880105871263718893999, 2.62548876893381550671462056413, 3.40532119917113448227826210075, 4.57856578128829407279366795195, 4.98711705259485142364187464826, 5.41256067580243381231398531207, 6.63984849171206317592199153186, 7.07849168213727086924731594792, 7.955847194745948757466401449731
