| L(s) = 1 | + 3·3-s − 19.3·5-s − 4.84·7-s + 9·9-s − 61.0·11-s − 13·13-s − 58.0·15-s − 41.7·17-s − 107.·19-s − 14.5·21-s − 28.5·23-s + 249.·25-s + 27·27-s + 89.8·29-s − 183.·31-s − 183.·33-s + 93.6·35-s − 418.·37-s − 39·39-s − 142.·41-s − 71.0·43-s − 174.·45-s − 323.·47-s − 319.·49-s − 125.·51-s + 25.1·53-s + 1.18e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.72·5-s − 0.261·7-s + 0.333·9-s − 1.67·11-s − 0.277·13-s − 0.998·15-s − 0.596·17-s − 1.29·19-s − 0.150·21-s − 0.258·23-s + 1.99·25-s + 0.192·27-s + 0.575·29-s − 1.06·31-s − 0.966·33-s + 0.452·35-s − 1.85·37-s − 0.160·39-s − 0.543·41-s − 0.252·43-s − 0.576·45-s − 1.00·47-s − 0.931·49-s − 0.344·51-s + 0.0650·53-s + 2.89·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.09933833069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09933833069\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 + 19.3T + 125T^{2} \) |
| 7 | \( 1 + 4.84T + 343T^{2} \) |
| 11 | \( 1 + 61.0T + 1.33e3T^{2} \) |
| 17 | \( 1 + 41.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 28.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 89.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 183.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 418.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 142.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 71.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 323.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 25.1T + 1.48e5T^{2} \) |
| 59 | \( 1 + 684.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 308.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 672.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 326.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 24.3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 166.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 201.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 108.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
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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
−8.311996479704367198012337777131, −7.997904767637554077588262595279, −7.21529788314404275732040974071, −6.54159351642870121880040582389, −5.15590519515866338054944612797, −4.53482624061084427885436829350, −3.63195102712601588450473728326, −2.97095673881482959713375786822, −1.94965332031322184640039832413, −0.12527939187395883882478731892,
0.12527939187395883882478731892, 1.94965332031322184640039832413, 2.97095673881482959713375786822, 3.63195102712601588450473728326, 4.53482624061084427885436829350, 5.15590519515866338054944612797, 6.54159351642870121880040582389, 7.21529788314404275732040974071, 7.997904767637554077588262595279, 8.311996479704367198012337777131
