| L(s) = 1 | − 1.89·2-s + 3.31·3-s + 1.58·4-s − 6.27·6-s − 7-s + 0.793·8-s + 7.99·9-s + 1.73·11-s + 5.24·12-s + 4.86·13-s + 1.89·14-s − 4.66·16-s + 0.684·17-s − 15.1·18-s − 4.52·19-s − 3.31·21-s − 3.28·22-s − 0.409·23-s + 2.63·24-s − 9.20·26-s + 16.5·27-s − 1.58·28-s − 8.84·29-s + 3.00·31-s + 7.23·32-s + 5.74·33-s − 1.29·34-s + ⋯ |
| L(s) = 1 | − 1.33·2-s + 1.91·3-s + 0.790·4-s − 2.56·6-s − 0.377·7-s + 0.280·8-s + 2.66·9-s + 0.522·11-s + 1.51·12-s + 1.34·13-s + 0.505·14-s − 1.16·16-s + 0.166·17-s − 3.56·18-s − 1.03·19-s − 0.723·21-s − 0.699·22-s − 0.0854·23-s + 0.536·24-s − 1.80·26-s + 3.18·27-s − 0.298·28-s − 1.64·29-s + 0.539·31-s + 1.27·32-s + 1.00·33-s − 0.222·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.629854053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.629854053\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + 1.89T + 2T^{2} \) |
| 3 | \( 1 - 3.31T + 3T^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 - 4.86T + 13T^{2} \) |
| 17 | \( 1 - 0.684T + 17T^{2} \) |
| 19 | \( 1 + 4.52T + 19T^{2} \) |
| 23 | \( 1 + 0.409T + 23T^{2} \) |
| 29 | \( 1 + 8.84T + 29T^{2} \) |
| 31 | \( 1 - 3.00T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 + 3.07T + 41T^{2} \) |
| 43 | \( 1 + 4.48T + 43T^{2} \) |
| 47 | \( 1 + 1.36T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 5.04T + 59T^{2} \) |
| 61 | \( 1 - 2.91T + 61T^{2} \) |
| 67 | \( 1 + 8.29T + 67T^{2} \) |
| 71 | \( 1 + 5.01T + 71T^{2} \) |
| 73 | \( 1 - 8.81T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 + 7.27T + 89T^{2} \) |
| 97 | \( 1 + 2.09T + 97T^{2} \) |
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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
−9.789487765697550324983926301503, −9.106625572921537620338266940731, −8.619473535051873920838627856315, −7.997718078802389247819441309735, −7.20721268574819677609524930633, −6.29324906178340266140727408465, −4.29962121915175203871230086709, −3.57077201997007894872516727635, −2.31207566929542967532956819648, −1.31121523482929626255462768630,
1.31121523482929626255462768630, 2.31207566929542967532956819648, 3.57077201997007894872516727635, 4.29962121915175203871230086709, 6.29324906178340266140727408465, 7.20721268574819677609524930633, 7.997718078802389247819441309735, 8.619473535051873920838627856315, 9.106625572921537620338266940731, 9.789487765697550324983926301503
