L(s) = 1 | − 0.732·3-s − 0.732·5-s − 7-s − 2.46·9-s + 1.46·11-s + 3.26·13-s + 0.535·15-s + 2·17-s − 4.73·19-s + 0.732·21-s − 3.46·23-s − 4.46·25-s + 4·27-s + 5.46·29-s + 4·31-s − 1.07·33-s + 0.732·35-s + 5.46·37-s − 2.39·39-s + 2·41-s + 1.46·43-s + 1.80·45-s − 10.9·47-s + 49-s − 1.46·51-s + 12·53-s − 1.07·55-s + ⋯ |
L(s) = 1 | − 0.422·3-s − 0.327·5-s − 0.377·7-s − 0.821·9-s + 0.441·11-s + 0.906·13-s + 0.138·15-s + 0.485·17-s − 1.08·19-s + 0.159·21-s − 0.722·23-s − 0.892·25-s + 0.769·27-s + 1.01·29-s + 0.718·31-s − 0.186·33-s + 0.123·35-s + 0.898·37-s − 0.383·39-s + 0.312·41-s + 0.223·43-s + 0.268·45-s − 1.59·47-s + 0.142·49-s − 0.205·51-s + 1.64·53-s − 0.144·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.147231032\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.147231032\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 5 | \( 1 + 0.732T + 5T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 - 3.26T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 4.73T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 5.46T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 5.46T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 1.46T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 + 7.66T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 0.928T + 73T^{2} \) |
| 79 | \( 1 - 2.92T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 + 4.92T + 97T^{2} \) |
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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
−9.253040624711519209680139610122, −8.360032357897011971093428836927, −7.942631667902056021715410311813, −6.52661247881130734402729411472, −6.27520771628208186617024231429, −5.32452588655350428160513412822, −4.21857438777618662389274167994, −3.46869413088191558390132097313, −2.30081170083472991406564070794, −0.73758995759133091522638295855,
0.73758995759133091522638295855, 2.30081170083472991406564070794, 3.46869413088191558390132097313, 4.21857438777618662389274167994, 5.32452588655350428160513412822, 6.27520771628208186617024231429, 6.52661247881130734402729411472, 7.942631667902056021715410311813, 8.360032357897011971093428836927, 9.253040624711519209680139610122