| L(s) = 1 | − 2.14·3-s + 2.09·5-s + 3.79·7-s + 1.58·9-s + 1.14·11-s + 1.19·13-s − 4.49·15-s + 0.537·17-s + 6.83·19-s − 8.11·21-s − 1.54·23-s − 0.595·25-s + 3.02·27-s + 2.80·29-s + 5.34·31-s − 2.44·33-s + 7.95·35-s + 0.325·37-s − 2.56·39-s + 11.9·41-s − 1.80·43-s + 3.32·45-s − 10.1·47-s + 7.37·49-s − 1.15·51-s − 10.0·53-s + 2.39·55-s + ⋯ |
| L(s) = 1 | − 1.23·3-s + 0.938·5-s + 1.43·7-s + 0.528·9-s + 0.344·11-s + 0.331·13-s − 1.16·15-s + 0.130·17-s + 1.56·19-s − 1.77·21-s − 0.322·23-s − 0.119·25-s + 0.582·27-s + 0.521·29-s + 0.959·31-s − 0.426·33-s + 1.34·35-s + 0.0535·37-s − 0.410·39-s + 1.86·41-s − 0.274·43-s + 0.496·45-s − 1.47·47-s + 1.05·49-s − 0.161·51-s − 1.38·53-s + 0.323·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.995468678\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.995468678\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + 2.14T + 3T^{2} \) |
| 5 | \( 1 - 2.09T + 5T^{2} \) |
| 7 | \( 1 - 3.79T + 7T^{2} \) |
| 11 | \( 1 - 1.14T + 11T^{2} \) |
| 13 | \( 1 - 1.19T + 13T^{2} \) |
| 17 | \( 1 - 0.537T + 17T^{2} \) |
| 19 | \( 1 - 6.83T + 19T^{2} \) |
| 23 | \( 1 + 1.54T + 23T^{2} \) |
| 29 | \( 1 - 2.80T + 29T^{2} \) |
| 31 | \( 1 - 5.34T + 31T^{2} \) |
| 37 | \( 1 - 0.325T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 + 1.80T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 7.07T + 61T^{2} \) |
| 67 | \( 1 + 6.68T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 - 4.74T + 73T^{2} \) |
| 79 | \( 1 - 0.815T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 + 1.94T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−8.256760649861475409707586176704, −7.80460945861232248582651444808, −6.71016041308650837603306953525, −6.14243208006721076077893490087, −5.37529648570150595395824274976, −5.02629031711172206568260986528, −4.13621598756205563598945433912, −2.81632169248530010965263712174, −1.64294382203801651587405611888, −0.956918196730973155619790176435,
0.956918196730973155619790176435, 1.64294382203801651587405611888, 2.81632169248530010965263712174, 4.13621598756205563598945433912, 5.02629031711172206568260986528, 5.37529648570150595395824274976, 6.14243208006721076077893490087, 6.71016041308650837603306953525, 7.80460945861232248582651444808, 8.256760649861475409707586176704
