| L(s) = 1 | + 3-s − 5-s + 2.78·7-s + 9-s + 4.92·11-s − 3.39·13-s − 15-s − 2.41·17-s + 6.92·19-s + 2.78·21-s + 3.80·23-s + 25-s + 27-s + 1.02·29-s − 31-s + 4.92·33-s − 2.78·35-s + 1.39·37-s − 3.39·39-s + 4·41-s + 1.38·43-s − 45-s + 2.37·47-s + 0.766·49-s − 2.41·51-s + 0.233·53-s − 4.92·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.05·7-s + 0.333·9-s + 1.48·11-s − 0.942·13-s − 0.258·15-s − 0.586·17-s + 1.58·19-s + 0.608·21-s + 0.793·23-s + 0.200·25-s + 0.192·27-s + 0.189·29-s − 0.179·31-s + 0.856·33-s − 0.471·35-s + 0.229·37-s − 0.543·39-s + 0.624·41-s + 0.211·43-s − 0.149·45-s + 0.346·47-s + 0.109·49-s − 0.338·51-s + 0.0321·53-s − 0.663·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.133176488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.133176488\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 2.78T + 7T^{2} \) |
| 11 | \( 1 - 4.92T + 11T^{2} \) |
| 13 | \( 1 + 3.39T + 13T^{2} \) |
| 17 | \( 1 + 2.41T + 17T^{2} \) |
| 19 | \( 1 - 6.92T + 19T^{2} \) |
| 23 | \( 1 - 3.80T + 23T^{2} \) |
| 29 | \( 1 - 1.02T + 29T^{2} \) |
| 37 | \( 1 - 1.39T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 1.38T + 43T^{2} \) |
| 47 | \( 1 - 2.37T + 47T^{2} \) |
| 53 | \( 1 - 0.233T + 53T^{2} \) |
| 59 | \( 1 + 3.29T + 59T^{2} \) |
| 61 | \( 1 + 4.31T + 61T^{2} \) |
| 67 | \( 1 + 0.176T + 67T^{2} \) |
| 71 | \( 1 + 3.58T + 71T^{2} \) |
| 73 | \( 1 - 2.74T + 73T^{2} \) |
| 79 | \( 1 + 5.06T + 79T^{2} \) |
| 83 | \( 1 - 5.63T + 83T^{2} \) |
| 89 | \( 1 - 9.94T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−7.84339319360957284926125096295, −7.29098513827389920487729850067, −6.77824910482243502334995911597, −5.73910781168905584231368354030, −4.83843594357390356478510918927, −4.39715860594401806443117503185, −3.54525175457924277954153308597, −2.74733592128433597633870629311, −1.73588337532180600710656249897, −0.930993008487317417094240965926,
0.930993008487317417094240965926, 1.73588337532180600710656249897, 2.74733592128433597633870629311, 3.54525175457924277954153308597, 4.39715860594401806443117503185, 4.83843594357390356478510918927, 5.73910781168905584231368354030, 6.77824910482243502334995911597, 7.29098513827389920487729850067, 7.84339319360957284926125096295
