| L(s) = 1 | − 0.493·2-s + 0.177·3-s − 1.75·4-s + 2.02·5-s − 0.0873·6-s − 2.21·7-s + 1.85·8-s − 2.96·9-s − 0.999·10-s + 1.54·11-s − 0.311·12-s + 13-s + 1.09·14-s + 0.358·15-s + 2.60·16-s + 0.982·17-s + 1.46·18-s − 2.38·19-s − 3.56·20-s − 0.392·21-s − 0.760·22-s + 1.84·23-s + 0.328·24-s − 0.893·25-s − 0.493·26-s − 1.05·27-s + 3.89·28-s + ⋯ |
| L(s) = 1 | − 0.348·2-s + 0.102·3-s − 0.878·4-s + 0.906·5-s − 0.0356·6-s − 0.837·7-s + 0.654·8-s − 0.989·9-s − 0.315·10-s + 0.465·11-s − 0.0898·12-s + 0.277·13-s + 0.291·14-s + 0.0926·15-s + 0.650·16-s + 0.238·17-s + 0.345·18-s − 0.548·19-s − 0.796·20-s − 0.0856·21-s − 0.162·22-s + 0.385·23-s + 0.0669·24-s − 0.178·25-s − 0.0967·26-s − 0.203·27-s + 0.735·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 0.493T + 2T^{2} \) |
| 3 | \( 1 - 0.177T + 3T^{2} \) |
| 5 | \( 1 - 2.02T + 5T^{2} \) |
| 7 | \( 1 + 2.21T + 7T^{2} \) |
| 11 | \( 1 - 1.54T + 11T^{2} \) |
| 17 | \( 1 - 0.982T + 17T^{2} \) |
| 19 | \( 1 + 2.38T + 19T^{2} \) |
| 23 | \( 1 - 1.84T + 23T^{2} \) |
| 29 | \( 1 - 2.52T + 29T^{2} \) |
| 31 | \( 1 + 8.33T + 31T^{2} \) |
| 37 | \( 1 + 6.86T + 37T^{2} \) |
| 41 | \( 1 - 2.96T + 41T^{2} \) |
| 43 | \( 1 + 1.87T + 43T^{2} \) |
| 47 | \( 1 - 1.03T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 - 5.66T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 6.00T + 67T^{2} \) |
| 71 | \( 1 - 7.95T + 71T^{2} \) |
| 73 | \( 1 + 16.6T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 5.89T + 97T^{2} \) |
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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
−9.144531213723383659454647329095, −8.745095624918712014402883913119, −7.77107177908326378779984833394, −6.62247548469610243274815394267, −5.86998921497265367041041995113, −5.14303923449618145974892593228, −3.91122300229773153529939128002, −3.01322845387493565925646255880, −1.62977128429294626684905435918, 0,
1.62977128429294626684905435918, 3.01322845387493565925646255880, 3.91122300229773153529939128002, 5.14303923449618145974892593228, 5.86998921497265367041041995113, 6.62247548469610243274815394267, 7.77107177908326378779984833394, 8.745095624918712014402883913119, 9.144531213723383659454647329095
