| L(s) = 1 | + 0.751·2-s − 3·3-s − 7.43·4-s − 3.91·5-s − 2.25·6-s + 9.58·7-s − 11.5·8-s + 9·9-s − 2.94·10-s + 11·11-s + 22.3·12-s + 52.7·13-s + 7.20·14-s + 11.7·15-s + 50.7·16-s + 124.·17-s + 6.76·18-s − 43.9·19-s + 29.1·20-s − 28.7·21-s + 8.26·22-s + 189.·23-s + 34.7·24-s − 109.·25-s + 39.6·26-s − 27·27-s − 71.2·28-s + ⋯ |
| L(s) = 1 | + 0.265·2-s − 0.577·3-s − 0.929·4-s − 0.350·5-s − 0.153·6-s + 0.517·7-s − 0.512·8-s + 0.333·9-s − 0.0929·10-s + 0.301·11-s + 0.536·12-s + 1.12·13-s + 0.137·14-s + 0.202·15-s + 0.793·16-s + 1.78·17-s + 0.0885·18-s − 0.530·19-s + 0.325·20-s − 0.298·21-s + 0.0800·22-s + 1.71·23-s + 0.295·24-s − 0.877·25-s + 0.298·26-s − 0.192·27-s − 0.481·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.719484479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.719484479\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 - 0.751T + 8T^{2} \) |
| 5 | \( 1 + 3.91T + 125T^{2} \) |
| 7 | \( 1 - 9.58T + 343T^{2} \) |
| 13 | \( 1 - 52.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 124.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 43.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 189.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 122.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 227.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 168.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 149.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 129.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 336.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 219.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 478.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 146.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 286.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 68.4T + 3.89e5T^{2} \) |
| 79 | \( 1 + 509.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.12e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 301.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 759.T + 9.12e5T^{2} \) |
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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
−8.707253568819967340781235603674, −8.133929113199297346790580790338, −7.24850065030553667662036079972, −6.23504601112615488440480432754, −5.42810436133579094517420958227, −4.88357096842176336627050656788, −3.84632268983195763070556448604, −3.31954601101008762020563906881, −1.49889118644261228659304876130, −0.66277518184656831319876386161,
0.66277518184656831319876386161, 1.49889118644261228659304876130, 3.31954601101008762020563906881, 3.84632268983195763070556448604, 4.88357096842176336627050656788, 5.42810436133579094517420958227, 6.23504601112615488440480432754, 7.24850065030553667662036079972, 8.133929113199297346790580790338, 8.707253568819967340781235603674
