| L(s) = 1 | − 1.33·2-s − 0.206·4-s + 0.887·5-s − 3.28·7-s + 2.95·8-s − 1.18·10-s + 3.91·11-s − 3.54·13-s + 4.40·14-s − 3.54·16-s − 3.14·17-s − 6.11·19-s − 0.183·20-s − 5.24·22-s + 23-s − 4.21·25-s + 4.75·26-s + 0.679·28-s − 29-s − 5.83·31-s − 1.16·32-s + 4.21·34-s − 2.91·35-s + 8.16·37-s + 8.18·38-s + 2.62·40-s − 6.59·41-s + ⋯ |
| L(s) = 1 | − 0.946·2-s − 0.103·4-s + 0.396·5-s − 1.24·7-s + 1.04·8-s − 0.375·10-s + 1.18·11-s − 0.983·13-s + 1.17·14-s − 0.885·16-s − 0.762·17-s − 1.40·19-s − 0.0410·20-s − 1.11·22-s + 0.208·23-s − 0.842·25-s + 0.931·26-s + 0.128·28-s − 0.185·29-s − 1.04·31-s − 0.205·32-s + 0.722·34-s − 0.493·35-s + 1.34·37-s + 1.32·38-s + 0.414·40-s − 1.02·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5131856597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5131856597\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.33T + 2T^{2} \) |
| 5 | \( 1 - 0.887T + 5T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 11 | \( 1 - 3.91T + 11T^{2} \) |
| 13 | \( 1 + 3.54T + 13T^{2} \) |
| 17 | \( 1 + 3.14T + 17T^{2} \) |
| 19 | \( 1 + 6.11T + 19T^{2} \) |
| 31 | \( 1 + 5.83T + 31T^{2} \) |
| 37 | \( 1 - 8.16T + 37T^{2} \) |
| 41 | \( 1 + 6.59T + 41T^{2} \) |
| 43 | \( 1 - 2.19T + 43T^{2} \) |
| 47 | \( 1 + 1.28T + 47T^{2} \) |
| 53 | \( 1 - 1.04T + 53T^{2} \) |
| 59 | \( 1 - 6.63T + 59T^{2} \) |
| 61 | \( 1 - 14.9T + 61T^{2} \) |
| 67 | \( 1 + 2.96T + 67T^{2} \) |
| 71 | \( 1 + 5.53T + 71T^{2} \) |
| 73 | \( 1 - 3.29T + 73T^{2} \) |
| 79 | \( 1 + 7.93T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 0.999T + 89T^{2} \) |
| 97 | \( 1 - 2.10T + 97T^{2} \) |
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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
−8.284778119967583728647605007000, −7.35194186546834374056035642656, −6.74114650257327817926550254329, −6.24052554318918216239480336370, −5.26580354107076630647100619038, −4.26596853955093873729450936560, −3.78068392146942733577962372084, −2.50220985514671496931407984329, −1.74760595575017804503328531465, −0.42747609730441551109174088454,
0.42747609730441551109174088454, 1.74760595575017804503328531465, 2.50220985514671496931407984329, 3.78068392146942733577962372084, 4.26596853955093873729450936560, 5.26580354107076630647100619038, 6.24052554318918216239480336370, 6.74114650257327817926550254329, 7.35194186546834374056035642656, 8.284778119967583728647605007000
