| L(s) = 1 | − 0.214·2-s − 1.95·4-s + 5-s − 3.65·7-s + 0.849·8-s − 0.214·10-s − 0.401·11-s − 13-s + 0.785·14-s + 3.72·16-s + 6.98·17-s + 3.87·19-s − 1.95·20-s + 0.0863·22-s + 5.80·23-s + 25-s + 0.214·26-s + 7.14·28-s − 7.63·29-s − 1.02·31-s − 2.49·32-s − 1.50·34-s − 3.65·35-s − 11.5·37-s − 0.831·38-s + 0.849·40-s − 10.1·41-s + ⋯ |
| L(s) = 1 | − 0.151·2-s − 0.976·4-s + 0.447·5-s − 1.38·7-s + 0.300·8-s − 0.0679·10-s − 0.121·11-s − 0.277·13-s + 0.209·14-s + 0.931·16-s + 1.69·17-s + 0.888·19-s − 0.436·20-s + 0.0183·22-s + 1.20·23-s + 0.200·25-s + 0.0421·26-s + 1.35·28-s − 1.41·29-s − 0.183·31-s − 0.441·32-s − 0.257·34-s − 0.618·35-s − 1.90·37-s − 0.134·38-s + 0.134·40-s − 1.58·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 0.214T + 2T^{2} \) |
| 7 | \( 1 + 3.65T + 7T^{2} \) |
| 11 | \( 1 + 0.401T + 11T^{2} \) |
| 17 | \( 1 - 6.98T + 17T^{2} \) |
| 19 | \( 1 - 3.87T + 19T^{2} \) |
| 23 | \( 1 - 5.80T + 23T^{2} \) |
| 29 | \( 1 + 7.63T + 29T^{2} \) |
| 31 | \( 1 + 1.02T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 2.87T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 1.68T + 59T^{2} \) |
| 61 | \( 1 - 4.49T + 61T^{2} \) |
| 67 | \( 1 + 7.59T + 67T^{2} \) |
| 71 | \( 1 - 7.86T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 + 9.31T + 83T^{2} \) |
| 89 | \( 1 + 3.29T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−9.141269385687189794937831928783, −8.219680969431266140309298306553, −7.33432775279541349521458415358, −6.54014487100151407121135114432, −5.40478609143125517917406650264, −5.10936769920525421232355646435, −3.46668464173583669302928080714, −3.27044802287274327542528958122, −1.44165086660122497131564555795, 0,
1.44165086660122497131564555795, 3.27044802287274327542528958122, 3.46668464173583669302928080714, 5.10936769920525421232355646435, 5.40478609143125517917406650264, 6.54014487100151407121135114432, 7.33432775279541349521458415358, 8.219680969431266140309298306553, 9.141269385687189794937831928783
