| L(s) = 1 | − 0.794·2-s + 0.0149·3-s − 1.36·4-s + 3.43·5-s − 0.0118·6-s + 2.82·7-s + 2.67·8-s − 2.99·9-s − 2.72·10-s + 3.50·11-s − 0.0204·12-s + 13-s − 2.24·14-s + 0.0512·15-s + 0.613·16-s + 5.15·17-s + 2.38·18-s − 1.66·19-s − 4.70·20-s + 0.0421·21-s − 2.78·22-s − 5.82·23-s + 0.0398·24-s + 6.80·25-s − 0.794·26-s − 0.0894·27-s − 3.86·28-s + ⋯ |
| L(s) = 1 | − 0.561·2-s + 0.00860·3-s − 0.684·4-s + 1.53·5-s − 0.00483·6-s + 1.06·7-s + 0.946·8-s − 0.999·9-s − 0.862·10-s + 1.05·11-s − 0.00589·12-s + 0.277·13-s − 0.599·14-s + 0.0132·15-s + 0.153·16-s + 1.25·17-s + 0.561·18-s − 0.381·19-s − 1.05·20-s + 0.00919·21-s − 0.593·22-s − 1.21·23-s + 0.00814·24-s + 1.36·25-s − 0.155·26-s − 0.0172·27-s − 0.731·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.545969265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.545969265\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 + 0.794T + 2T^{2} \) |
| 3 | \( 1 - 0.0149T + 3T^{2} \) |
| 5 | \( 1 - 3.43T + 5T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 - 3.50T + 11T^{2} \) |
| 17 | \( 1 - 5.15T + 17T^{2} \) |
| 19 | \( 1 + 1.66T + 19T^{2} \) |
| 23 | \( 1 + 5.82T + 23T^{2} \) |
| 29 | \( 1 + 3.06T + 29T^{2} \) |
| 31 | \( 1 - 7.05T + 31T^{2} \) |
| 37 | \( 1 + 2.44T + 37T^{2} \) |
| 41 | \( 1 + 6.63T + 41T^{2} \) |
| 43 | \( 1 - 5.70T + 43T^{2} \) |
| 47 | \( 1 + 5.15T + 47T^{2} \) |
| 53 | \( 1 - 8.82T + 53T^{2} \) |
| 59 | \( 1 - 2.57T + 59T^{2} \) |
| 61 | \( 1 - 6.11T + 61T^{2} \) |
| 67 | \( 1 + 9.16T + 67T^{2} \) |
| 71 | \( 1 - 5.67T + 71T^{2} \) |
| 73 | \( 1 - 2.78T + 73T^{2} \) |
| 83 | \( 1 - 3.49T + 83T^{2} \) |
| 89 | \( 1 - 9.94T + 89T^{2} \) |
| 97 | \( 1 - 4.45T + 97T^{2} \) |
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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
−9.914807314455678460173095796698, −9.065555207403652051836608986576, −8.496774766071909363127789527712, −7.76546191736361770223195877339, −6.38067443479027787856245822735, −5.64666910925615446578259963327, −4.90745351324680341604897387030, −3.69850553222556603099007784631, −2.11114940150595537439901642562, −1.18258613286613865458031054073,
1.18258613286613865458031054073, 2.11114940150595537439901642562, 3.69850553222556603099007784631, 4.90745351324680341604897387030, 5.64666910925615446578259963327, 6.38067443479027787856245822735, 7.76546191736361770223195877339, 8.496774766071909363127789527712, 9.065555207403652051836608986576, 9.914807314455678460173095796698
