| L(s) = 1 | − 2.55·2-s − 3.16·3-s + 4.53·4-s − 0.685·5-s + 8.09·6-s + 1.59·7-s − 6.48·8-s + 7.01·9-s + 1.75·10-s + 5.60·11-s − 14.3·12-s + 6.89·13-s − 4.07·14-s + 2.17·15-s + 7.51·16-s − 4.01·17-s − 17.9·18-s − 0.308·19-s − 3.11·20-s − 5.04·21-s − 14.3·22-s + 1.61·23-s + 20.5·24-s − 4.52·25-s − 17.6·26-s − 12.7·27-s + 7.23·28-s + ⋯ |
| L(s) = 1 | − 1.80·2-s − 1.82·3-s + 2.26·4-s − 0.306·5-s + 3.30·6-s + 0.602·7-s − 2.29·8-s + 2.33·9-s + 0.554·10-s + 1.68·11-s − 4.14·12-s + 1.91·13-s − 1.08·14-s + 0.560·15-s + 1.87·16-s − 0.973·17-s − 4.22·18-s − 0.0708·19-s − 0.695·20-s − 1.10·21-s − 3.05·22-s + 0.337·23-s + 4.19·24-s − 0.905·25-s − 3.45·26-s − 2.44·27-s + 1.36·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5414682978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5414682978\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 8011 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 3 | \( 1 + 3.16T + 3T^{2} \) |
| 5 | \( 1 + 0.685T + 5T^{2} \) |
| 7 | \( 1 - 1.59T + 7T^{2} \) |
| 11 | \( 1 - 5.60T + 11T^{2} \) |
| 13 | \( 1 - 6.89T + 13T^{2} \) |
| 17 | \( 1 + 4.01T + 17T^{2} \) |
| 19 | \( 1 + 0.308T + 19T^{2} \) |
| 23 | \( 1 - 1.61T + 23T^{2} \) |
| 29 | \( 1 - 7.76T + 29T^{2} \) |
| 31 | \( 1 + 0.885T + 31T^{2} \) |
| 37 | \( 1 + 8.39T + 37T^{2} \) |
| 41 | \( 1 - 6.70T + 41T^{2} \) |
| 43 | \( 1 + 9.78T + 43T^{2} \) |
| 47 | \( 1 - 1.40T + 47T^{2} \) |
| 53 | \( 1 - 0.875T + 53T^{2} \) |
| 59 | \( 1 + 1.30T + 59T^{2} \) |
| 61 | \( 1 - 5.98T + 61T^{2} \) |
| 67 | \( 1 - 0.601T + 67T^{2} \) |
| 71 | \( 1 - 8.18T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 - 7.71T + 79T^{2} \) |
| 83 | \( 1 - 9.63T + 83T^{2} \) |
| 89 | \( 1 + 0.465T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−8.018162061500417923089060846603, −6.88654824175120016952312075605, −6.61646671954231247234008895652, −6.23599766928888747399919038297, −5.29820748163594368597887091877, −4.31181763063676585142901175997, −3.62691321295156771792000722610, −1.87121477492033567572768688306, −1.28969390210755265154251059528, −0.62802860636241365284242885560,
0.62802860636241365284242885560, 1.28969390210755265154251059528, 1.87121477492033567572768688306, 3.62691321295156771792000722610, 4.31181763063676585142901175997, 5.29820748163594368597887091877, 6.23599766928888747399919038297, 6.61646671954231247234008895652, 6.88654824175120016952312075605, 8.018162061500417923089060846603
