| L(s) = 1 | − 1.27·2-s − 3-s − 0.372·4-s − 0.506·5-s + 1.27·6-s − 3.05·7-s + 3.02·8-s + 9-s + 0.645·10-s − 1.77·11-s + 0.372·12-s + 13-s + 3.89·14-s + 0.506·15-s − 3.11·16-s − 3.59·17-s − 1.27·18-s − 0.715·19-s + 0.188·20-s + 3.05·21-s + 2.26·22-s − 0.238·23-s − 3.02·24-s − 4.74·25-s − 1.27·26-s − 27-s + 1.13·28-s + ⋯ |
| L(s) = 1 | − 0.902·2-s − 0.577·3-s − 0.186·4-s − 0.226·5-s + 0.520·6-s − 1.15·7-s + 1.07·8-s + 0.333·9-s + 0.204·10-s − 0.535·11-s + 0.107·12-s + 0.277·13-s + 1.04·14-s + 0.130·15-s − 0.779·16-s − 0.871·17-s − 0.300·18-s − 0.164·19-s + 0.0421·20-s + 0.666·21-s + 0.483·22-s − 0.0497·23-s − 0.617·24-s − 0.948·25-s − 0.250·26-s − 0.192·27-s + 0.214·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2288658312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2288658312\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 1.27T + 2T^{2} \) |
| 5 | \( 1 + 0.506T + 5T^{2} \) |
| 7 | \( 1 + 3.05T + 7T^{2} \) |
| 11 | \( 1 + 1.77T + 11T^{2} \) |
| 17 | \( 1 + 3.59T + 17T^{2} \) |
| 19 | \( 1 + 0.715T + 19T^{2} \) |
| 23 | \( 1 + 0.238T + 23T^{2} \) |
| 29 | \( 1 - 2.95T + 29T^{2} \) |
| 31 | \( 1 - 0.136T + 31T^{2} \) |
| 37 | \( 1 - 5.90T + 37T^{2} \) |
| 41 | \( 1 + 5.81T + 41T^{2} \) |
| 43 | \( 1 + 2.36T + 43T^{2} \) |
| 47 | \( 1 + 7.27T + 47T^{2} \) |
| 53 | \( 1 + 4.12T + 53T^{2} \) |
| 59 | \( 1 + 4.06T + 59T^{2} \) |
| 61 | \( 1 + 3.05T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 - 6.06T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 - 1.65T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−8.406688426484965562826417906441, −7.910875312910079008056334614521, −6.95590392802035352787287590824, −6.44743406373552587319281208290, −5.56067504498333197314954569273, −4.64122085390894371817764036366, −3.94323090499832305255747826075, −2.88669602465494498735075058343, −1.64556809164748647723040584888, −0.32392210636250452958421956677,
0.32392210636250452958421956677, 1.64556809164748647723040584888, 2.88669602465494498735075058343, 3.94323090499832305255747826075, 4.64122085390894371817764036366, 5.56067504498333197314954569273, 6.44743406373552587319281208290, 6.95590392802035352787287590824, 7.910875312910079008056334614521, 8.406688426484965562826417906441
