| L(s) = 1 | + 2-s − 2.97·3-s + 4-s − 2.48·5-s − 2.97·6-s + 4.88·7-s + 8-s + 5.86·9-s − 2.48·10-s + 1.65·11-s − 2.97·12-s + 4.91·13-s + 4.88·14-s + 7.38·15-s + 16-s + 4.32·17-s + 5.86·18-s + 1.98·19-s − 2.48·20-s − 14.5·21-s + 1.65·22-s + 4.62·23-s − 2.97·24-s + 1.15·25-s + 4.91·26-s − 8.53·27-s + 4.88·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.71·3-s + 0.5·4-s − 1.10·5-s − 1.21·6-s + 1.84·7-s + 0.353·8-s + 1.95·9-s − 0.784·10-s + 0.500·11-s − 0.859·12-s + 1.36·13-s + 1.30·14-s + 1.90·15-s + 0.250·16-s + 1.04·17-s + 1.38·18-s + 0.455·19-s − 0.554·20-s − 3.17·21-s + 0.353·22-s + 0.963·23-s − 0.607·24-s + 0.231·25-s + 0.963·26-s − 1.64·27-s + 0.923·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.142882380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.142882380\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 + 2.97T + 3T^{2} \) |
| 5 | \( 1 + 2.48T + 5T^{2} \) |
| 7 | \( 1 - 4.88T + 7T^{2} \) |
| 11 | \( 1 - 1.65T + 11T^{2} \) |
| 13 | \( 1 - 4.91T + 13T^{2} \) |
| 17 | \( 1 - 4.32T + 17T^{2} \) |
| 19 | \( 1 - 1.98T + 19T^{2} \) |
| 23 | \( 1 - 4.62T + 23T^{2} \) |
| 29 | \( 1 + 1.48T + 29T^{2} \) |
| 31 | \( 1 - 0.164T + 31T^{2} \) |
| 37 | \( 1 - 0.0499T + 37T^{2} \) |
| 41 | \( 1 - 2.74T + 41T^{2} \) |
| 43 | \( 1 + 0.530T + 43T^{2} \) |
| 47 | \( 1 - 7.52T + 47T^{2} \) |
| 53 | \( 1 + 9.24T + 53T^{2} \) |
| 59 | \( 1 + 8.61T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 + 7.60T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 - 4.66T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 0.564T + 97T^{2} \) |
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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
−8.124349803554387230346321233348, −7.54821258916689557079108572828, −6.93174304066483396447172945738, −5.87820638037092431980281118692, −5.52278249203531976617748500834, −4.62768316604306176636787867142, −4.25525414385578648075490242489, −3.34786753871613131547870954102, −1.53069831562929514321825401867, −0.954895765040442698268305273894,
0.954895765040442698268305273894, 1.53069831562929514321825401867, 3.34786753871613131547870954102, 4.25525414385578648075490242489, 4.62768316604306176636787867142, 5.52278249203531976617748500834, 5.87820638037092431980281118692, 6.93174304066483396447172945738, 7.54821258916689557079108572828, 8.124349803554387230346321233348
