| L(s) = 1 | − 1.38·2-s + 3-s − 0.0927·4-s − 0.547·5-s − 1.38·6-s + 1.24·7-s + 2.89·8-s + 9-s + 0.756·10-s + 11-s − 0.0927·12-s + 4.55·13-s − 1.71·14-s − 0.547·15-s − 3.80·16-s − 0.308·17-s − 1.38·18-s − 3.94·19-s + 0.0508·20-s + 1.24·21-s − 1.38·22-s + 8.68·23-s + 2.89·24-s − 4.69·25-s − 6.28·26-s + 27-s − 0.115·28-s + ⋯ |
| L(s) = 1 | − 0.976·2-s + 0.577·3-s − 0.0463·4-s − 0.245·5-s − 0.563·6-s + 0.469·7-s + 1.02·8-s + 0.333·9-s + 0.239·10-s + 0.301·11-s − 0.0267·12-s + 1.26·13-s − 0.458·14-s − 0.141·15-s − 0.951·16-s − 0.0747·17-s − 0.325·18-s − 0.904·19-s + 0.0113·20-s + 0.270·21-s − 0.294·22-s + 1.81·23-s + 0.589·24-s − 0.939·25-s − 1.23·26-s + 0.192·27-s − 0.0217·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.304957972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.304957972\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 1.38T + 2T^{2} \) |
| 5 | \( 1 + 0.547T + 5T^{2} \) |
| 7 | \( 1 - 1.24T + 7T^{2} \) |
| 13 | \( 1 - 4.55T + 13T^{2} \) |
| 17 | \( 1 + 0.308T + 17T^{2} \) |
| 19 | \( 1 + 3.94T + 19T^{2} \) |
| 23 | \( 1 - 8.68T + 23T^{2} \) |
| 29 | \( 1 - 1.88T + 29T^{2} \) |
| 31 | \( 1 - 6.18T + 31T^{2} \) |
| 37 | \( 1 - 3.67T + 37T^{2} \) |
| 41 | \( 1 - 2.00T + 41T^{2} \) |
| 43 | \( 1 + 5.76T + 43T^{2} \) |
| 47 | \( 1 + 8.17T + 47T^{2} \) |
| 53 | \( 1 - 5.53T + 53T^{2} \) |
| 59 | \( 1 + 1.48T + 59T^{2} \) |
| 67 | \( 1 + 7.33T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 6.36T + 73T^{2} \) |
| 79 | \( 1 - 4.39T + 79T^{2} \) |
| 83 | \( 1 - 8.36T + 83T^{2} \) |
| 89 | \( 1 - 2.09T + 89T^{2} \) |
| 97 | \( 1 - 0.697T + 97T^{2} \) |
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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
−9.019911039304504156646853961629, −8.359498006926009272043039675041, −8.034812789902949193900451924925, −7.03359534867808545334083775641, −6.25319677873899038850051646370, −4.91326861147850507713470328979, −4.22327195869817705476713758753, −3.24757972012084752820187727262, −1.88633082495173156649583972627, −0.919027056118635817466288094606,
0.919027056118635817466288094606, 1.88633082495173156649583972627, 3.24757972012084752820187727262, 4.22327195869817705476713758753, 4.91326861147850507713470328979, 6.25319677873899038850051646370, 7.03359534867808545334083775641, 8.034812789902949193900451924925, 8.359498006926009272043039675041, 9.019911039304504156646853961629
