| L(s) = 1 | − 1.31·2-s + 1.87·3-s − 0.268·4-s − 1.28·5-s − 2.46·6-s − 2.16·7-s + 2.98·8-s + 0.512·9-s + 1.69·10-s + 1.34·11-s − 0.502·12-s + 5.76·13-s + 2.84·14-s − 2.40·15-s − 3.39·16-s + 3.88·17-s − 0.674·18-s + 8.19·19-s + 0.344·20-s − 4.05·21-s − 1.77·22-s + 5.98·23-s + 5.59·24-s − 3.35·25-s − 7.59·26-s − 4.66·27-s + 0.580·28-s + ⋯ |
| L(s) = 1 | − 0.930·2-s + 1.08·3-s − 0.134·4-s − 0.574·5-s − 1.00·6-s − 0.817·7-s + 1.05·8-s + 0.170·9-s + 0.534·10-s + 0.407·11-s − 0.145·12-s + 1.59·13-s + 0.761·14-s − 0.621·15-s − 0.847·16-s + 0.942·17-s − 0.158·18-s + 1.88·19-s + 0.0770·20-s − 0.885·21-s − 0.378·22-s + 1.24·23-s + 1.14·24-s − 0.670·25-s − 1.48·26-s − 0.897·27-s + 0.109·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 409 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 409 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.015043951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.015043951\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 409 | \( 1 - T \) |
| good | 2 | \( 1 + 1.31T + 2T^{2} \) |
| 3 | \( 1 - 1.87T + 3T^{2} \) |
| 5 | \( 1 + 1.28T + 5T^{2} \) |
| 7 | \( 1 + 2.16T + 7T^{2} \) |
| 11 | \( 1 - 1.34T + 11T^{2} \) |
| 13 | \( 1 - 5.76T + 13T^{2} \) |
| 17 | \( 1 - 3.88T + 17T^{2} \) |
| 19 | \( 1 - 8.19T + 19T^{2} \) |
| 23 | \( 1 - 5.98T + 23T^{2} \) |
| 29 | \( 1 - 4.67T + 29T^{2} \) |
| 31 | \( 1 + 4.78T + 31T^{2} \) |
| 37 | \( 1 - 1.37T + 37T^{2} \) |
| 41 | \( 1 + 6.84T + 41T^{2} \) |
| 43 | \( 1 + 2.99T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 3.69T + 53T^{2} \) |
| 59 | \( 1 + 2.10T + 59T^{2} \) |
| 61 | \( 1 - 9.36T + 61T^{2} \) |
| 67 | \( 1 + 4.89T + 67T^{2} \) |
| 71 | \( 1 - 8.93T + 71T^{2} \) |
| 73 | \( 1 + 2.75T + 73T^{2} \) |
| 79 | \( 1 + 16.4T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 + 6.38T + 89T^{2} \) |
| 97 | \( 1 + 1.98T + 97T^{2} \) |
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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
−11.05040910796081948391078192809, −9.935212303050378380853105787149, −9.267378560690608868639632288385, −8.606568859660852374623602128930, −7.85869835174669895241756467497, −7.00763048236790694359369427962, −5.52061922678777277893232850090, −3.82346990727214686351441567333, −3.19355407316583390992572537534, −1.15559528276882229191020674156,
1.15559528276882229191020674156, 3.19355407316583390992572537534, 3.82346990727214686351441567333, 5.52061922678777277893232850090, 7.00763048236790694359369427962, 7.85869835174669895241756467497, 8.606568859660852374623602128930, 9.267378560690608868639632288385, 9.935212303050378380853105787149, 11.05040910796081948391078192809
