| L(s) = 1 | − 3-s − 0.864·5-s − 7-s + 9-s + 3.52·11-s + 4·13-s + 0.864·15-s − 0.864·17-s − 19-s + 21-s + 3.52·23-s − 4.25·25-s − 27-s − 4.38·29-s + 1.52·31-s − 3.52·33-s + 0.864·35-s + 2·37-s − 4·39-s + 10.7·41-s − 5.52·43-s − 0.864·45-s − 6.38·47-s + 49-s + 0.864·51-s − 1.34·53-s − 3.04·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.386·5-s − 0.377·7-s + 0.333·9-s + 1.06·11-s + 1.10·13-s + 0.223·15-s − 0.209·17-s − 0.229·19-s + 0.218·21-s + 0.734·23-s − 0.850·25-s − 0.192·27-s − 0.814·29-s + 0.273·31-s − 0.613·33-s + 0.146·35-s + 0.328·37-s − 0.640·39-s + 1.68·41-s − 0.842·43-s − 0.128·45-s − 0.931·47-s + 0.142·49-s + 0.121·51-s − 0.184·53-s − 0.410·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.386604787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.386604787\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 0.864T + 5T^{2} \) |
| 11 | \( 1 - 3.52T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 0.864T + 17T^{2} \) |
| 23 | \( 1 - 3.52T + 23T^{2} \) |
| 29 | \( 1 + 4.38T + 29T^{2} \) |
| 31 | \( 1 - 1.52T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 5.52T + 43T^{2} \) |
| 47 | \( 1 + 6.38T + 47T^{2} \) |
| 53 | \( 1 + 1.34T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 + 3.52T + 67T^{2} \) |
| 71 | \( 1 + 4.11T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 + 5.25T + 79T^{2} \) |
| 83 | \( 1 - 8.11T + 83T^{2} \) |
| 89 | \( 1 + 6.77T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−8.744468297372415125513352639630, −7.88317942641693012488783809610, −7.06296056686310212041801342550, −6.31199051359271210012986130471, −5.86627081615606313954444557162, −4.73925092924410617821929570878, −3.95587404589981668193774263159, −3.30757105671798526735996101808, −1.86302393491333755449469799399, −0.74904572332696076940314921049,
0.74904572332696076940314921049, 1.86302393491333755449469799399, 3.30757105671798526735996101808, 3.95587404589981668193774263159, 4.73925092924410617821929570878, 5.86627081615606313954444557162, 6.31199051359271210012986130471, 7.06296056686310212041801342550, 7.88317942641693012488783809610, 8.744468297372415125513352639630
