| L(s) = 1 | − 2-s + 4-s − 0.732·7-s − 8-s + 5.19·11-s + 1.26·13-s + 0.732·14-s + 16-s − 0.732·17-s + 19-s − 5.19·22-s − 1.53·23-s − 1.26·26-s − 0.732·28-s − 1.19·29-s + 7.92·31-s − 32-s + 0.732·34-s + 4.92·37-s − 38-s + 5.26·43-s + 5.19·44-s + 1.53·46-s − 3.46·47-s − 6.46·49-s + 1.26·52-s − 12.6·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.276·7-s − 0.353·8-s + 1.56·11-s + 0.351·13-s + 0.195·14-s + 0.250·16-s − 0.177·17-s + 0.229·19-s − 1.10·22-s − 0.320·23-s − 0.248·26-s − 0.138·28-s − 0.222·29-s + 1.42·31-s − 0.176·32-s + 0.125·34-s + 0.810·37-s − 0.162·38-s + 0.803·43-s + 0.783·44-s + 0.226·46-s − 0.505·47-s − 0.923·49-s + 0.175·52-s − 1.73·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.631908525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.631908525\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 0.732T + 7T^{2} \) |
| 11 | \( 1 - 5.19T + 11T^{2} \) |
| 13 | \( 1 - 1.26T + 13T^{2} \) |
| 17 | \( 1 + 0.732T + 17T^{2} \) |
| 23 | \( 1 + 1.53T + 23T^{2} \) |
| 29 | \( 1 + 1.19T + 29T^{2} \) |
| 31 | \( 1 - 7.92T + 31T^{2} \) |
| 37 | \( 1 - 4.92T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 5.26T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 - 8.19T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 5.19T + 67T^{2} \) |
| 71 | \( 1 + 0.196T + 71T^{2} \) |
| 73 | \( 1 - 7.53T + 73T^{2} \) |
| 79 | \( 1 + 7.92T + 79T^{2} \) |
| 83 | \( 1 - 0.660T + 83T^{2} \) |
| 89 | \( 1 - 9.92T + 89T^{2} \) |
| 97 | \( 1 + 7.12T + 97T^{2} \) |
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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
−7.911064003222787022762921733525, −7.06034308734641872097516305304, −6.40907172957973903251332362444, −6.10224920022176731277633195519, −4.99750819007311893243617994920, −4.11363323690529548395805094837, −3.47238015038960897478556419937, −2.53172827065228540030614164639, −1.54705923089067991151376419143, −0.74033170388853075284429841489,
0.74033170388853075284429841489, 1.54705923089067991151376419143, 2.53172827065228540030614164639, 3.47238015038960897478556419937, 4.11363323690529548395805094837, 4.99750819007311893243617994920, 6.10224920022176731277633195519, 6.40907172957973903251332362444, 7.06034308734641872097516305304, 7.911064003222787022762921733525
