| L(s) = 1 | + 2-s + 3.23·3-s + 4-s + 4.14·5-s + 3.23·6-s − 0.0175·7-s + 8-s + 7.48·9-s + 4.14·10-s − 5.76·11-s + 3.23·12-s − 0.223·13-s − 0.0175·14-s + 13.4·15-s + 16-s − 7.03·17-s + 7.48·18-s + 7.18·19-s + 4.14·20-s − 0.0567·21-s − 5.76·22-s + 23-s + 3.23·24-s + 12.1·25-s − 0.223·26-s + 14.5·27-s − 0.0175·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.86·3-s + 0.5·4-s + 1.85·5-s + 1.32·6-s − 0.00662·7-s + 0.353·8-s + 2.49·9-s + 1.31·10-s − 1.73·11-s + 0.934·12-s − 0.0619·13-s − 0.00468·14-s + 3.46·15-s + 0.250·16-s − 1.70·17-s + 1.76·18-s + 1.64·19-s + 0.927·20-s − 0.0123·21-s − 1.22·22-s + 0.208·23-s + 0.661·24-s + 2.43·25-s − 0.0438·26-s + 2.79·27-s − 0.00331·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.647195307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.647195307\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 3.23T + 3T^{2} \) |
| 5 | \( 1 - 4.14T + 5T^{2} \) |
| 7 | \( 1 + 0.0175T + 7T^{2} \) |
| 11 | \( 1 + 5.76T + 11T^{2} \) |
| 13 | \( 1 + 0.223T + 13T^{2} \) |
| 17 | \( 1 + 7.03T + 17T^{2} \) |
| 19 | \( 1 - 7.18T + 19T^{2} \) |
| 29 | \( 1 - 6.84T + 29T^{2} \) |
| 31 | \( 1 + 5.98T + 31T^{2} \) |
| 37 | \( 1 + 2.31T + 37T^{2} \) |
| 41 | \( 1 - 8.20T + 41T^{2} \) |
| 43 | \( 1 + 4.56T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 - 4.59T + 53T^{2} \) |
| 59 | \( 1 + 3.79T + 59T^{2} \) |
| 61 | \( 1 + 6.15T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 - 0.976T + 71T^{2} \) |
| 73 | \( 1 - 7.68T + 73T^{2} \) |
| 79 | \( 1 - 2.70T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 - 9.18T + 97T^{2} \) |
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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
−8.112142115800511700256570604829, −7.33735593955430258346399996766, −6.76109597097252506149911883839, −5.86703922750707565035805665190, −5.02041103794315542005872123374, −4.57823262483192897419578823350, −3.17838532780244024754936217604, −2.83926447493470574089599624483, −2.16836805968500479230561418084, −1.52152448045850422364505599948,
1.52152448045850422364505599948, 2.16836805968500479230561418084, 2.83926447493470574089599624483, 3.17838532780244024754936217604, 4.57823262483192897419578823350, 5.02041103794315542005872123374, 5.86703922750707565035805665190, 6.76109597097252506149911883839, 7.33735593955430258346399996766, 8.112142115800511700256570604829
