| L(s) = 1 | − 1.73·2-s − 5·4-s − 1.73·5-s − 13.8·7-s + 22.5·8-s + 2.99·10-s + 13.8·11-s + 23.9·14-s + 1.00·16-s + 117·17-s + 114.·19-s + 8.66·20-s − 23.9·22-s − 78·23-s − 122·25-s + 69.2·28-s + 141·29-s − 155.·31-s − 181.·32-s − 202.·34-s + 23.9·35-s + 143.·37-s − 198·38-s − 38.9·40-s − 271.·41-s − 104·43-s − 69.2·44-s + ⋯ |
| L(s) = 1 | − 0.612·2-s − 0.625·4-s − 0.154·5-s − 0.748·7-s + 0.995·8-s + 0.0948·10-s + 0.379·11-s + 0.458·14-s + 0.0156·16-s + 1.66·17-s + 1.38·19-s + 0.0968·20-s − 0.232·22-s − 0.707·23-s − 0.975·25-s + 0.467·28-s + 0.902·29-s − 0.903·31-s − 1.00·32-s − 1.02·34-s + 0.115·35-s + 0.638·37-s − 0.845·38-s − 0.154·40-s − 1.03·41-s − 0.368·43-s − 0.237·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.003731872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.003731872\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.73T + 8T^{2} \) |
| 5 | \( 1 + 1.73T + 125T^{2} \) |
| 7 | \( 1 + 13.8T + 343T^{2} \) |
| 11 | \( 1 - 13.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 117T + 4.91e3T^{2} \) |
| 19 | \( 1 - 114.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 78T + 1.21e4T^{2} \) |
| 29 | \( 1 - 141T + 2.43e4T^{2} \) |
| 31 | \( 1 + 155.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 143.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 271.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 104T + 7.95e4T^{2} \) |
| 47 | \( 1 + 301.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 93T + 1.48e5T^{2} \) |
| 59 | \( 1 - 284.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 145T + 2.26e5T^{2} \) |
| 67 | \( 1 + 786.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.05e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 458.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.27e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 789.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 976.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 200.T + 9.12e5T^{2} \) |
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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
−9.327486932291249285724095862797, −8.206063465790956996325082526317, −7.77423134477657699827200500138, −6.83742857468351378204704595839, −5.77759699241434142784358207487, −5.02172039165984458568670852084, −3.82991968795437775432555786804, −3.22096572160771695183392359924, −1.57727971356795834644781448684, −0.57034227746350611523180720085,
0.57034227746350611523180720085, 1.57727971356795834644781448684, 3.22096572160771695183392359924, 3.82991968795437775432555786804, 5.02172039165984458568670852084, 5.77759699241434142784358207487, 6.83742857468351378204704595839, 7.77423134477657699827200500138, 8.206063465790956996325082526317, 9.327486932291249285724095862797
