| L(s) = 1 | − 1.61·2-s + 3-s + 0.618·4-s − 1.23·5-s − 1.61·6-s + 2.23·8-s − 2·9-s + 2.00·10-s − 4.47·11-s + 0.618·12-s + 4.23·13-s − 1.23·15-s − 4.85·16-s + 3.23·18-s + 2.76·19-s − 0.763·20-s + 7.23·22-s − 23-s + 2.23·24-s − 3.47·25-s − 6.85·26-s − 5·27-s + 7.47·29-s + 2.00·30-s + 9·31-s + 3.38·32-s − 4.47·33-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.577·3-s + 0.309·4-s − 0.552·5-s − 0.660·6-s + 0.790·8-s − 0.666·9-s + 0.632·10-s − 1.34·11-s + 0.178·12-s + 1.17·13-s − 0.319·15-s − 1.21·16-s + 0.762·18-s + 0.634·19-s − 0.170·20-s + 1.54·22-s − 0.208·23-s + 0.456·24-s − 0.694·25-s − 1.34·26-s − 0.962·27-s + 1.38·29-s + 0.365·30-s + 1.61·31-s + 0.597·32-s − 0.778·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7570098606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7570098606\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 13 | \( 1 - 4.23T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 29 | \( 1 - 7.47T + 29T^{2} \) |
| 31 | \( 1 - 9T + 31T^{2} \) |
| 37 | \( 1 - 7.70T + 37T^{2} \) |
| 41 | \( 1 + 2.23T + 41T^{2} \) |
| 43 | \( 1 + 6.47T + 43T^{2} \) |
| 47 | \( 1 + 5.47T + 47T^{2} \) |
| 53 | \( 1 - 6.76T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 + 5.76T + 71T^{2} \) |
| 73 | \( 1 + 6.70T + 73T^{2} \) |
| 79 | \( 1 + 2.76T + 79T^{2} \) |
| 83 | \( 1 - 2.47T + 83T^{2} \) |
| 89 | \( 1 - 8.94T + 89T^{2} \) |
| 97 | \( 1 - 9.70T + 97T^{2} \) |
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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
−9.860379609929391644596808189317, −8.745639440852546702786070108862, −8.138474069919238163754920597156, −7.991131461537774865548882492784, −6.79856321375055198010680089203, −5.63520962271459585262950663416, −4.54363451617843676591953035776, −3.39132672155244407036368218894, −2.35284789215438961085117919597, −0.75480863510713876683498937908,
0.75480863510713876683498937908, 2.35284789215438961085117919597, 3.39132672155244407036368218894, 4.54363451617843676591953035776, 5.63520962271459585262950663416, 6.79856321375055198010680089203, 7.991131461537774865548882492784, 8.138474069919238163754920597156, 8.745639440852546702786070108862, 9.860379609929391644596808189317
