| L(s) = 1 | − 2.66·2-s + 1.08·3-s + 5.11·4-s − 1.31·5-s − 2.88·6-s + 1.98·7-s − 8.30·8-s − 1.82·9-s + 3.50·10-s − 5.31·11-s + 5.53·12-s − 13-s − 5.28·14-s − 1.42·15-s + 11.9·16-s − 3.62·17-s + 4.87·18-s − 2.73·19-s − 6.71·20-s + 2.14·21-s + 14.1·22-s + 4.66·23-s − 8.98·24-s − 3.27·25-s + 2.66·26-s − 5.22·27-s + 10.1·28-s + ⋯ |
| L(s) = 1 | − 1.88·2-s + 0.625·3-s + 2.55·4-s − 0.587·5-s − 1.17·6-s + 0.749·7-s − 2.93·8-s − 0.609·9-s + 1.10·10-s − 1.60·11-s + 1.59·12-s − 0.277·13-s − 1.41·14-s − 0.367·15-s + 2.97·16-s − 0.879·17-s + 1.14·18-s − 0.627·19-s − 1.50·20-s + 0.468·21-s + 3.02·22-s + 0.971·23-s − 1.83·24-s − 0.654·25-s + 0.523·26-s − 1.00·27-s + 1.91·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 2.66T + 2T^{2} \) |
| 3 | \( 1 - 1.08T + 3T^{2} \) |
| 5 | \( 1 + 1.31T + 5T^{2} \) |
| 7 | \( 1 - 1.98T + 7T^{2} \) |
| 11 | \( 1 + 5.31T + 11T^{2} \) |
| 17 | \( 1 + 3.62T + 17T^{2} \) |
| 19 | \( 1 + 2.73T + 19T^{2} \) |
| 23 | \( 1 - 4.66T + 23T^{2} \) |
| 29 | \( 1 - 8.58T + 29T^{2} \) |
| 37 | \( 1 + 8.71T + 37T^{2} \) |
| 41 | \( 1 - 7.10T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 0.147T + 53T^{2} \) |
| 59 | \( 1 - 2.12T + 59T^{2} \) |
| 61 | \( 1 + 7.00T + 61T^{2} \) |
| 67 | \( 1 - 5.05T + 67T^{2} \) |
| 71 | \( 1 + 1.02T + 71T^{2} \) |
| 73 | \( 1 - 5.21T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + 4.93T + 83T^{2} \) |
| 89 | \( 1 - 1.77T + 89T^{2} \) |
| 97 | \( 1 + 7.08T + 97T^{2} \) |
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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
−10.74181232855903459856486719022, −9.777784183154288329936165631629, −8.659587677921334878455780179664, −8.253237459802799919605659723883, −7.64364593402998125485674982704, −6.58632453427176673736568570945, −5.04454236254919929545568131099, −3.02480756303301781897269614583, −2.06326340777272610218749326388, 0,
2.06326340777272610218749326388, 3.02480756303301781897269614583, 5.04454236254919929545568131099, 6.58632453427176673736568570945, 7.64364593402998125485674982704, 8.253237459802799919605659723883, 8.659587677921334878455780179664, 9.777784183154288329936165631629, 10.74181232855903459856486719022
