| L(s) = 1 | + 0.732·3-s − 5-s − 7-s − 2.46·9-s − 11-s − 2.73·13-s − 0.732·15-s + 1.26·17-s − 5.46·19-s − 0.732·21-s + 25-s − 4·27-s − 3.46·29-s + 6.19·31-s − 0.732·33-s + 35-s + 5.46·37-s − 2·39-s + 8.19·41-s − 2·43-s + 2.46·45-s − 11.6·47-s + 49-s + 0.928·51-s + 9.46·53-s + 55-s − 4·57-s + ⋯ |
| L(s) = 1 | + 0.422·3-s − 0.447·5-s − 0.377·7-s − 0.821·9-s − 0.301·11-s − 0.757·13-s − 0.189·15-s + 0.307·17-s − 1.25·19-s − 0.159·21-s + 0.200·25-s − 0.769·27-s − 0.643·29-s + 1.11·31-s − 0.127·33-s + 0.169·35-s + 0.898·37-s − 0.320·39-s + 1.28·41-s − 0.304·43-s + 0.367·45-s − 1.70·47-s + 0.142·49-s + 0.129·51-s + 1.29·53-s + 0.134·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.185334137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.185334137\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 17 | \( 1 - 1.26T + 17T^{2} \) |
| 19 | \( 1 + 5.46T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 - 6.19T + 31T^{2} \) |
| 37 | \( 1 - 5.46T + 37T^{2} \) |
| 41 | \( 1 - 8.19T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 9.46T + 53T^{2} \) |
| 59 | \( 1 + 7.26T + 59T^{2} \) |
| 61 | \( 1 + 5.26T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 9.46T + 71T^{2} \) |
| 73 | \( 1 + 0.196T + 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 - 4.39T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−7.944098393639419172248466848208, −7.62698924601171132001851968067, −6.57545462224965720969424159669, −6.05141868713886652223417446497, −5.12485392150681259097271238815, −4.39206021325138839795414404190, −3.53378487177113829499156399885, −2.77538881175124599528562246040, −2.10103471827852538456931982174, −0.52531431900004226826379122031,
0.52531431900004226826379122031, 2.10103471827852538456931982174, 2.77538881175124599528562246040, 3.53378487177113829499156399885, 4.39206021325138839795414404190, 5.12485392150681259097271238815, 6.05141868713886652223417446497, 6.57545462224965720969424159669, 7.62698924601171132001851968067, 7.944098393639419172248466848208
