| L(s) = 1 | + 2-s + 2.87·3-s + 4-s − 5-s + 2.87·6-s + 7-s + 8-s + 5.24·9-s − 10-s + 3.57·11-s + 2.87·12-s − 5.82·13-s + 14-s − 2.87·15-s + 16-s + 17-s + 5.24·18-s + 1.49·19-s − 20-s + 2.87·21-s + 3.57·22-s + 1.79·23-s + 2.87·24-s + 25-s − 5.82·26-s + 6.44·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.65·3-s + 0.5·4-s − 0.447·5-s + 1.17·6-s + 0.377·7-s + 0.353·8-s + 1.74·9-s − 0.316·10-s + 1.07·11-s + 0.828·12-s − 1.61·13-s + 0.267·14-s − 0.741·15-s + 0.250·16-s + 0.242·17-s + 1.23·18-s + 0.343·19-s − 0.223·20-s + 0.626·21-s + 0.762·22-s + 0.374·23-s + 0.586·24-s + 0.200·25-s − 1.14·26-s + 1.24·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.300446474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.300446474\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 2.87T + 3T^{2} \) |
| 11 | \( 1 - 3.57T + 11T^{2} \) |
| 13 | \( 1 + 5.82T + 13T^{2} \) |
| 19 | \( 1 - 1.49T + 19T^{2} \) |
| 23 | \( 1 - 1.79T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 + 4.65T + 31T^{2} \) |
| 37 | \( 1 - 9.15T + 37T^{2} \) |
| 41 | \( 1 + 7.31T + 41T^{2} \) |
| 43 | \( 1 + 3.54T + 43T^{2} \) |
| 47 | \( 1 + 8.61T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 6.02T + 59T^{2} \) |
| 61 | \( 1 + 8.41T + 61T^{2} \) |
| 67 | \( 1 + 5.31T + 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 + 7.07T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 - 9.30T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−9.523188959015038076715367548896, −8.998115656325536214620217947501, −7.933027583681415172793057923537, −7.46221445999968961835252791928, −6.68099932871645482458360066568, −5.22102260530054301459941989858, −4.32357034768631235051266159607, −3.54916329606979422882413236528, −2.68724760503198968132719484004, −1.66033948874283463927320799532,
1.66033948874283463927320799532, 2.68724760503198968132719484004, 3.54916329606979422882413236528, 4.32357034768631235051266159607, 5.22102260530054301459941989858, 6.68099932871645482458360066568, 7.46221445999968961835252791928, 7.933027583681415172793057923537, 8.998115656325536214620217947501, 9.523188959015038076715367548896
