| L(s) = 1 | + 3-s − 4.21·5-s + 7-s + 9-s − 5.57·11-s − 1.35·13-s − 4.21·15-s − 4.86·17-s + 2.64·19-s + 21-s − 2.86·23-s + 12.7·25-s + 27-s − 2.64·29-s − 1.35·31-s − 5.57·33-s − 4.21·35-s − 5.79·37-s − 1.35·39-s + 0.425·41-s + 8.43·43-s − 4.21·45-s + 11.1·47-s + 49-s − 4.86·51-s − 5.79·53-s + 23.5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.88·5-s + 0.377·7-s + 0.333·9-s − 1.68·11-s − 0.376·13-s − 1.08·15-s − 1.17·17-s + 0.606·19-s + 0.218·21-s − 0.596·23-s + 2.55·25-s + 0.192·27-s − 0.490·29-s − 0.243·31-s − 0.970·33-s − 0.713·35-s − 0.952·37-s − 0.217·39-s + 0.0663·41-s + 1.28·43-s − 0.628·45-s + 1.62·47-s + 0.142·49-s − 0.680·51-s − 0.795·53-s + 3.17·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8697474046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8697474046\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 4.21T + 5T^{2} \) |
| 11 | \( 1 + 5.57T + 11T^{2} \) |
| 13 | \( 1 + 1.35T + 13T^{2} \) |
| 17 | \( 1 + 4.86T + 17T^{2} \) |
| 19 | \( 1 - 2.64T + 19T^{2} \) |
| 23 | \( 1 + 2.86T + 23T^{2} \) |
| 29 | \( 1 + 2.64T + 29T^{2} \) |
| 31 | \( 1 + 1.35T + 31T^{2} \) |
| 37 | \( 1 + 5.79T + 37T^{2} \) |
| 41 | \( 1 - 0.425T + 41T^{2} \) |
| 43 | \( 1 - 8.43T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 5.79T + 53T^{2} \) |
| 59 | \( 1 + 9.72T + 59T^{2} \) |
| 61 | \( 1 + 15.0T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 - 1.56T + 73T^{2} \) |
| 79 | \( 1 - 0.436T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 - 9.14T + 97T^{2} \) |
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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
−7.997133171861454623564215165468, −7.49072714701430459571510315956, −7.31383728113386947948984938495, −6.03217638130432095758600373888, −4.91335883820452824524937189607, −4.53740689263379891385435165636, −3.65534848475923715054724500159, −2.94397358118513562923130696792, −2.08339744454559113826202527035, −0.46154861675791770967768661077,
0.46154861675791770967768661077, 2.08339744454559113826202527035, 2.94397358118513562923130696792, 3.65534848475923715054724500159, 4.53740689263379891385435165636, 4.91335883820452824524937189607, 6.03217638130432095758600373888, 7.31383728113386947948984938495, 7.49072714701430459571510315956, 7.997133171861454623564215165468
