| L(s) = 1 | − 3·3-s − 4.51·5-s + 7.48·7-s + 9·9-s − 66.8·11-s + 13·13-s + 13.5·15-s + 96.9·17-s + 31.4·19-s − 22.4·21-s − 183.·23-s − 104.·25-s − 27·27-s − 112.·29-s + 77.2·31-s + 200.·33-s − 33.7·35-s − 54.7·37-s − 39·39-s + 451.·41-s − 113.·43-s − 40.6·45-s + 42.2·47-s − 287·49-s − 290.·51-s + 530.·53-s + 302.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.403·5-s + 0.404·7-s + 0.333·9-s − 1.83·11-s + 0.277·13-s + 0.233·15-s + 1.38·17-s + 0.380·19-s − 0.233·21-s − 1.66·23-s − 0.836·25-s − 0.192·27-s − 0.718·29-s + 0.447·31-s + 1.05·33-s − 0.163·35-s − 0.243·37-s − 0.160·39-s + 1.72·41-s − 0.402·43-s − 0.134·45-s + 0.131·47-s − 0.836·49-s − 0.798·51-s + 1.37·53-s + 0.740·55-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
| \(L(2)\) |
\(\approx\) |
\(0.9750847727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9750847727\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 4.51T + 125T^{2} \) |
| 7 | \( 1 - 7.48T + 343T^{2} \) |
| 11 | \( 1 + 66.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 96.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 31.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 183.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 112.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 77.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 54.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 451.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 113.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 42.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 530.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 219.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 822.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 872.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 100.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 165.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 545.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 454.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 230.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.08e3T + 9.12e5T^{2} \) |
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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
−8.214749650847019865179813266684, −7.83311175650407435717480886285, −7.29609653330291740539923491394, −5.87304165299979100831131316746, −5.65230746249493606781413848497, −4.68904464806183213011307753558, −3.81951254332740743580081873021, −2.79407484006407047849924043753, −1.69210227831788071392018475629, −0.44383129657166711667091732709,
0.44383129657166711667091732709, 1.69210227831788071392018475629, 2.79407484006407047849924043753, 3.81951254332740743580081873021, 4.68904464806183213011307753558, 5.65230746249493606781413848497, 5.87304165299979100831131316746, 7.29609653330291740539923491394, 7.83311175650407435717480886285, 8.214749650847019865179813266684
