| L(s) = 1 | + 1.20·3-s + 0.863·5-s + 7-s − 1.54·9-s + 0.0275·11-s − 0.319·13-s + 1.04·15-s − 17-s − 1.03·19-s + 1.20·21-s − 8.22·23-s − 4.25·25-s − 5.48·27-s − 1.97·29-s − 8.12·31-s + 0.0332·33-s + 0.863·35-s − 8.86·37-s − 0.385·39-s − 3.61·41-s − 5.96·43-s − 1.33·45-s + 8.25·47-s + 49-s − 1.20·51-s + 8.14·53-s + 0.0237·55-s + ⋯ |
| L(s) = 1 | + 0.696·3-s + 0.386·5-s + 0.377·7-s − 0.514·9-s + 0.00829·11-s − 0.0886·13-s + 0.269·15-s − 0.242·17-s − 0.236·19-s + 0.263·21-s − 1.71·23-s − 0.850·25-s − 1.05·27-s − 0.366·29-s − 1.45·31-s + 0.00578·33-s + 0.146·35-s − 1.45·37-s − 0.0617·39-s − 0.564·41-s − 0.909·43-s − 0.198·45-s + 1.20·47-s + 0.142·49-s − 0.168·51-s + 1.11·53-s + 0.00320·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 1.20T + 3T^{2} \) |
| 5 | \( 1 - 0.863T + 5T^{2} \) |
| 11 | \( 1 - 0.0275T + 11T^{2} \) |
| 13 | \( 1 + 0.319T + 13T^{2} \) |
| 19 | \( 1 + 1.03T + 19T^{2} \) |
| 23 | \( 1 + 8.22T + 23T^{2} \) |
| 29 | \( 1 + 1.97T + 29T^{2} \) |
| 31 | \( 1 + 8.12T + 31T^{2} \) |
| 37 | \( 1 + 8.86T + 37T^{2} \) |
| 41 | \( 1 + 3.61T + 41T^{2} \) |
| 43 | \( 1 + 5.96T + 43T^{2} \) |
| 47 | \( 1 - 8.25T + 47T^{2} \) |
| 53 | \( 1 - 8.14T + 53T^{2} \) |
| 59 | \( 1 - 7.34T + 59T^{2} \) |
| 61 | \( 1 + 0.736T + 61T^{2} \) |
| 67 | \( 1 - 0.540T + 67T^{2} \) |
| 71 | \( 1 + 1.76T + 71T^{2} \) |
| 73 | \( 1 + 3.55T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 7.27T + 83T^{2} \) |
| 89 | \( 1 + 7.62T + 89T^{2} \) |
| 97 | \( 1 - 5.01T + 97T^{2} \) |
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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
−8.188222878533690242293226323710, −7.56922055359131016245976426404, −6.70439539605040165657893416155, −5.75098633546280971857541954630, −5.30069729039936896268101564347, −4.05801637100844488657570708476, −3.51128721028254343729343367741, −2.28602680729985539429309963429, −1.83320613593143170744321438761, 0,
1.83320613593143170744321438761, 2.28602680729985539429309963429, 3.51128721028254343729343367741, 4.05801637100844488657570708476, 5.30069729039936896268101564347, 5.75098633546280971857541954630, 6.70439539605040165657893416155, 7.56922055359131016245976426404, 8.188222878533690242293226323710
