L(s) = 1 | + 2-s − 0.0566·3-s + 4-s + 5-s − 0.0566·6-s − 5.00·7-s + 8-s − 2.99·9-s + 10-s − 2.22·11-s − 0.0566·12-s − 2.99·13-s − 5.00·14-s − 0.0566·15-s + 16-s − 1.17·17-s − 2.99·18-s + 7.27·19-s + 20-s + 0.283·21-s − 2.22·22-s − 0.0566·24-s + 25-s − 2.99·26-s + 0.339·27-s − 5.00·28-s + 2.78·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.0327·3-s + 0.5·4-s + 0.447·5-s − 0.0231·6-s − 1.89·7-s + 0.353·8-s − 0.998·9-s + 0.316·10-s − 0.671·11-s − 0.0163·12-s − 0.831·13-s − 1.33·14-s − 0.0146·15-s + 0.250·16-s − 0.284·17-s − 0.706·18-s + 1.66·19-s + 0.223·20-s + 0.0619·21-s − 0.474·22-s − 0.0115·24-s + 0.200·25-s − 0.587·26-s + 0.0654·27-s − 0.946·28-s + 0.518·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.921388206\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.921388206\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + 0.0566T + 3T^{2} \) |
| 7 | \( 1 + 5.00T + 7T^{2} \) |
| 11 | \( 1 + 2.22T + 11T^{2} \) |
| 13 | \( 1 + 2.99T + 13T^{2} \) |
| 17 | \( 1 + 1.17T + 17T^{2} \) |
| 19 | \( 1 - 7.27T + 19T^{2} \) |
| 29 | \( 1 - 2.78T + 29T^{2} \) |
| 31 | \( 1 - 7.90T + 31T^{2} \) |
| 37 | \( 1 + 0.498T + 37T^{2} \) |
| 41 | \( 1 - 5.54T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 4.15T + 47T^{2} \) |
| 53 | \( 1 + 2.72T + 53T^{2} \) |
| 59 | \( 1 + 1.61T + 59T^{2} \) |
| 61 | \( 1 - 5.58T + 61T^{2} \) |
| 67 | \( 1 + 6.27T + 67T^{2} \) |
| 71 | \( 1 - 9.80T + 71T^{2} \) |
| 73 | \( 1 - 1.68T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 - 1.37T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 - 4.93T + 97T^{2} \) |
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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
−8.088503554133923204222426077570, −7.24236810426730790057483308649, −6.60164618176017008669981802513, −5.97627470211340788822547670645, −5.39007250297232563200595218134, −4.65494801979866630030721037978, −3.40565647007408945282349375257, −2.96439095903651092701215434875, −2.36940544289619169407687904743, −0.63253857403941753658103427183,
0.63253857403941753658103427183, 2.36940544289619169407687904743, 2.96439095903651092701215434875, 3.40565647007408945282349375257, 4.65494801979866630030721037978, 5.39007250297232563200595218134, 5.97627470211340788822547670645, 6.60164618176017008669981802513, 7.24236810426730790057483308649, 8.088503554133923204222426077570