L(s) = 1 | − 2.31·2-s − 2.66·3-s + 3.37·4-s + 6.19·6-s + 0.0358·7-s − 3.20·8-s + 4.12·9-s + 3.56·11-s − 9.02·12-s − 0.958·13-s − 0.0832·14-s + 0.663·16-s − 6.20·17-s − 9.56·18-s − 4.21·19-s − 0.0958·21-s − 8.26·22-s − 6.00·23-s + 8.54·24-s + 2.22·26-s − 3.00·27-s + 0.121·28-s − 7.87·29-s − 6.60·31-s + 4.86·32-s − 9.51·33-s + 14.4·34-s + ⋯ |
L(s) = 1 | − 1.64·2-s − 1.54·3-s + 1.68·4-s + 2.52·6-s + 0.0135·7-s − 1.13·8-s + 1.37·9-s + 1.07·11-s − 2.60·12-s − 0.265·13-s − 0.0222·14-s + 0.165·16-s − 1.50·17-s − 2.25·18-s − 0.965·19-s − 0.0209·21-s − 1.76·22-s − 1.25·23-s + 1.74·24-s + 0.436·26-s − 0.577·27-s + 0.0229·28-s − 1.46·29-s − 1.18·31-s + 0.859·32-s − 1.65·33-s + 2.46·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.05674334439\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05674334439\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
good | 2 | \( 1 + 2.31T + 2T^{2} \) |
| 3 | \( 1 + 2.66T + 3T^{2} \) |
| 7 | \( 1 - 0.0358T + 7T^{2} \) |
| 11 | \( 1 - 3.56T + 11T^{2} \) |
| 13 | \( 1 + 0.958T + 13T^{2} \) |
| 17 | \( 1 + 6.20T + 17T^{2} \) |
| 19 | \( 1 + 4.21T + 19T^{2} \) |
| 23 | \( 1 + 6.00T + 23T^{2} \) |
| 29 | \( 1 + 7.87T + 29T^{2} \) |
| 31 | \( 1 + 6.60T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + 3.51T + 41T^{2} \) |
| 43 | \( 1 - 1.12T + 43T^{2} \) |
| 47 | \( 1 + 8.54T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 - 7.49T + 61T^{2} \) |
| 67 | \( 1 + 5.18T + 67T^{2} \) |
| 71 | \( 1 - 7.29T + 71T^{2} \) |
| 73 | \( 1 - 4.01T + 73T^{2} \) |
| 79 | \( 1 + 3.76T + 79T^{2} \) |
| 83 | \( 1 + 0.267T + 83T^{2} \) |
| 89 | \( 1 - 5.03T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−8.209250535867450030847521664647, −7.22536920169137123671295521322, −6.74676076824209007882888156015, −6.29565707034057712014912713144, −5.46278368411007312879535537456, −4.53403039868103105037818639665, −3.76187578194827787225682813094, −2.06224569790646136430798985794, −1.58840762102914574138132768025, −0.17433585059051036150325853841,
0.17433585059051036150325853841, 1.58840762102914574138132768025, 2.06224569790646136430798985794, 3.76187578194827787225682813094, 4.53403039868103105037818639665, 5.46278368411007312879535537456, 6.29565707034057712014912713144, 6.74676076824209007882888156015, 7.22536920169137123671295521322, 8.209250535867450030847521664647