| L(s) = 1 | − 2.31·2-s + 2.93·3-s + 3.33·4-s + 2.06·5-s − 6.77·6-s − 0.615·7-s − 3.08·8-s + 5.60·9-s − 4.77·10-s + 4.10·11-s + 9.78·12-s + 2.91·13-s + 1.42·14-s + 6.06·15-s + 0.460·16-s + 7.15·17-s − 12.9·18-s − 4.02·19-s + 6.90·20-s − 1.80·21-s − 9.49·22-s + 2.67·23-s − 9.05·24-s − 0.721·25-s − 6.73·26-s + 7.64·27-s − 2.05·28-s + ⋯ |
| L(s) = 1 | − 1.63·2-s + 1.69·3-s + 1.66·4-s + 0.925·5-s − 2.76·6-s − 0.232·7-s − 1.09·8-s + 1.86·9-s − 1.51·10-s + 1.23·11-s + 2.82·12-s + 0.808·13-s + 0.379·14-s + 1.56·15-s + 0.115·16-s + 1.73·17-s − 3.05·18-s − 0.924·19-s + 1.54·20-s − 0.393·21-s − 2.02·22-s + 0.557·23-s − 1.84·24-s − 0.144·25-s − 1.32·26-s + 1.47·27-s − 0.387·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.654395618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.654395618\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 53 | \( 1 + T \) |
| 151 | \( 1 - T \) |
| good | 2 | \( 1 + 2.31T + 2T^{2} \) |
| 3 | \( 1 - 2.93T + 3T^{2} \) |
| 5 | \( 1 - 2.06T + 5T^{2} \) |
| 7 | \( 1 + 0.615T + 7T^{2} \) |
| 11 | \( 1 - 4.10T + 11T^{2} \) |
| 13 | \( 1 - 2.91T + 13T^{2} \) |
| 17 | \( 1 - 7.15T + 17T^{2} \) |
| 19 | \( 1 + 4.02T + 19T^{2} \) |
| 23 | \( 1 - 2.67T + 23T^{2} \) |
| 29 | \( 1 + 2.92T + 29T^{2} \) |
| 31 | \( 1 - 4.57T + 31T^{2} \) |
| 37 | \( 1 + 7.07T + 37T^{2} \) |
| 41 | \( 1 + 5.98T + 41T^{2} \) |
| 43 | \( 1 + 8.52T + 43T^{2} \) |
| 47 | \( 1 - 3.52T + 47T^{2} \) |
| 59 | \( 1 - 8.19T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 3.56T + 67T^{2} \) |
| 71 | \( 1 + 3.32T + 71T^{2} \) |
| 73 | \( 1 + 3.68T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 - 18.6T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−8.163806299443465992635408281358, −7.46667424910310546101135025063, −6.66673167350960059058632610735, −6.28211489743720828127381184659, −5.08384773369686783673881402111, −3.74556864817235901455662381315, −3.37220901484883577180727916120, −2.27348865475273923572833218752, −1.69864620957208095218378761579, −1.04710339142231282684539851849,
1.04710339142231282684539851849, 1.69864620957208095218378761579, 2.27348865475273923572833218752, 3.37220901484883577180727916120, 3.74556864817235901455662381315, 5.08384773369686783673881402111, 6.28211489743720828127381184659, 6.66673167350960059058632610735, 7.46667424910310546101135025063, 8.163806299443465992635408281358
