| L(s) = 1 | − 1.93·2-s − 1.02·3-s + 1.72·4-s + 3.42·5-s + 1.97·6-s − 3.76·7-s + 0.521·8-s − 1.95·9-s − 6.60·10-s + 11-s − 1.76·12-s − 3.64·13-s + 7.27·14-s − 3.49·15-s − 4.46·16-s − 17-s + 3.77·18-s − 5.82·19-s + 5.91·20-s + 3.85·21-s − 1.93·22-s + 6.80·23-s − 0.532·24-s + 6.69·25-s + 7.04·26-s + 5.06·27-s − 6.52·28-s + ⋯ |
| L(s) = 1 | − 1.36·2-s − 0.589·3-s + 0.864·4-s + 1.52·5-s + 0.805·6-s − 1.42·7-s + 0.184·8-s − 0.652·9-s − 2.08·10-s + 0.301·11-s − 0.510·12-s − 1.01·13-s + 1.94·14-s − 0.902·15-s − 1.11·16-s − 0.242·17-s + 0.890·18-s − 1.33·19-s + 1.32·20-s + 0.840·21-s − 0.411·22-s + 1.41·23-s − 0.108·24-s + 1.33·25-s + 1.38·26-s + 0.974·27-s − 1.23·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5022555550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5022555550\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 1.93T + 2T^{2} \) |
| 3 | \( 1 + 1.02T + 3T^{2} \) |
| 5 | \( 1 - 3.42T + 5T^{2} \) |
| 7 | \( 1 + 3.76T + 7T^{2} \) |
| 13 | \( 1 + 3.64T + 13T^{2} \) |
| 19 | \( 1 + 5.82T + 19T^{2} \) |
| 23 | \( 1 - 6.80T + 23T^{2} \) |
| 29 | \( 1 - 8.99T + 29T^{2} \) |
| 31 | \( 1 - 8.93T + 31T^{2} \) |
| 37 | \( 1 - 5.86T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 47 | \( 1 + 8.53T + 47T^{2} \) |
| 53 | \( 1 + 1.76T + 53T^{2} \) |
| 59 | \( 1 + 8.40T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 6.02T + 71T^{2} \) |
| 73 | \( 1 + 9.17T + 73T^{2} \) |
| 79 | \( 1 + 6.50T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + 6.66T + 89T^{2} \) |
| 97 | \( 1 - 2.21T + 97T^{2} \) |
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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
−8.074072878037867481257707202252, −6.84427132065616446838685697606, −6.47984642984855016024442732909, −6.25649003110640087972826112972, −5.08747795488042178662332333095, −4.60711427903228165265292844103, −2.95386733671361110207548018336, −2.58234317476650909628765483307, −1.49196001296926371503875065363, −0.45330927788907692924964108316,
0.45330927788907692924964108316, 1.49196001296926371503875065363, 2.58234317476650909628765483307, 2.95386733671361110207548018336, 4.60711427903228165265292844103, 5.08747795488042178662332333095, 6.25649003110640087972826112972, 6.47984642984855016024442732909, 6.84427132065616446838685697606, 8.074072878037867481257707202252
