| L(s) = 1 | + 1.46·2-s − 3-s + 0.139·4-s + 3.72·5-s − 1.46·6-s − 1.32·7-s − 2.72·8-s + 9-s + 5.44·10-s − 5.32·11-s − 0.139·12-s + 2.39·13-s − 1.93·14-s − 3.72·15-s − 4.25·16-s − 5.04·17-s + 1.46·18-s + 0.925·19-s + 0.518·20-s + 1.32·21-s − 7.78·22-s + 5.72·23-s + 2.72·24-s + 8.85·25-s + 3.50·26-s − 27-s − 0.184·28-s + ⋯ |
| L(s) = 1 | + 1.03·2-s − 0.577·3-s + 0.0695·4-s + 1.66·5-s − 0.597·6-s − 0.500·7-s − 0.962·8-s + 0.333·9-s + 1.72·10-s − 1.60·11-s − 0.0401·12-s + 0.665·13-s − 0.517·14-s − 0.960·15-s − 1.06·16-s − 1.22·17-s + 0.344·18-s + 0.212·19-s + 0.115·20-s + 0.288·21-s − 1.65·22-s + 1.19·23-s + 0.555·24-s + 1.77·25-s + 0.687·26-s − 0.192·27-s − 0.0348·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.353708713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.353708713\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.46T + 2T^{2} \) |
| 5 | \( 1 - 3.72T + 5T^{2} \) |
| 7 | \( 1 + 1.32T + 7T^{2} \) |
| 11 | \( 1 + 5.32T + 11T^{2} \) |
| 13 | \( 1 - 2.39T + 13T^{2} \) |
| 17 | \( 1 + 5.04T + 17T^{2} \) |
| 19 | \( 1 - 0.925T + 19T^{2} \) |
| 23 | \( 1 - 5.72T + 23T^{2} \) |
| 31 | \( 1 - 5.72T + 31T^{2} \) |
| 37 | \( 1 - 1.07T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 9.29T + 43T^{2} \) |
| 47 | \( 1 - 3.97T + 47T^{2} \) |
| 53 | \( 1 - 2.12T + 53T^{2} \) |
| 59 | \( 1 + 1.35T + 59T^{2} \) |
| 61 | \( 1 - 2.92T + 61T^{2} \) |
| 67 | \( 1 + 7.97T + 67T^{2} \) |
| 71 | \( 1 + 4.12T + 71T^{2} \) |
| 73 | \( 1 + 1.07T + 73T^{2} \) |
| 79 | \( 1 - 8.36T + 79T^{2} \) |
| 83 | \( 1 + 4.79T + 83T^{2} \) |
| 89 | \( 1 + 0.547T + 89T^{2} \) |
| 97 | \( 1 + 5.57T + 97T^{2} \) |
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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
−13.55279162438137608597014146268, −13.44892495713606964181803530730, −12.56347159316272050538248874774, −10.97150262426378419697287322230, −9.980046809237475541709886375772, −8.845876907000719840823775694028, −6.62921567822898276532781437031, −5.72754420404654707034576687104, −4.84531528171055805336140400292, −2.73387108539709217330115468964,
2.73387108539709217330115468964, 4.84531528171055805336140400292, 5.72754420404654707034576687104, 6.62921567822898276532781437031, 8.845876907000719840823775694028, 9.980046809237475541709886375772, 10.97150262426378419697287322230, 12.56347159316272050538248874774, 13.44892495713606964181803530730, 13.55279162438137608597014146268
