L(s) = 1 | + 0.294·2-s + 3-s − 1.91·4-s + 0.386·5-s + 0.294·6-s + 2.95·7-s − 1.15·8-s + 9-s + 0.113·10-s − 1.64·11-s − 1.91·12-s + 2.41·13-s + 0.871·14-s + 0.386·15-s + 3.48·16-s + 1.61·17-s + 0.294·18-s + 4.40·19-s − 0.739·20-s + 2.95·21-s − 0.486·22-s + 6.60·23-s − 1.15·24-s − 4.85·25-s + 0.712·26-s + 27-s − 5.65·28-s + ⋯ |
L(s) = 1 | + 0.208·2-s + 0.577·3-s − 0.956·4-s + 0.172·5-s + 0.120·6-s + 1.11·7-s − 0.407·8-s + 0.333·9-s + 0.0360·10-s − 0.497·11-s − 0.552·12-s + 0.670·13-s + 0.232·14-s + 0.0998·15-s + 0.871·16-s + 0.392·17-s + 0.0694·18-s + 1.01·19-s − 0.165·20-s + 0.645·21-s − 0.103·22-s + 1.37·23-s − 0.235·24-s − 0.970·25-s + 0.139·26-s + 0.192·27-s − 1.06·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.868037843\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.868037843\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 193 | \( 1 + T \) |
good | 2 | \( 1 - 0.294T + 2T^{2} \) |
| 5 | \( 1 - 0.386T + 5T^{2} \) |
| 7 | \( 1 - 2.95T + 7T^{2} \) |
| 11 | \( 1 + 1.64T + 11T^{2} \) |
| 13 | \( 1 - 2.41T + 13T^{2} \) |
| 17 | \( 1 - 1.61T + 17T^{2} \) |
| 19 | \( 1 - 4.40T + 19T^{2} \) |
| 23 | \( 1 - 6.60T + 23T^{2} \) |
| 29 | \( 1 + 1.93T + 29T^{2} \) |
| 31 | \( 1 + 4.51T + 31T^{2} \) |
| 37 | \( 1 - 5.95T + 37T^{2} \) |
| 41 | \( 1 - 5.29T + 41T^{2} \) |
| 43 | \( 1 - 8.10T + 43T^{2} \) |
| 47 | \( 1 - 0.468T + 47T^{2} \) |
| 53 | \( 1 + 13.5T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 8.10T + 61T^{2} \) |
| 67 | \( 1 + 1.47T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 - 4.78T + 73T^{2} \) |
| 79 | \( 1 + 6.29T + 79T^{2} \) |
| 83 | \( 1 + 4.71T + 83T^{2} \) |
| 89 | \( 1 + 3.18T + 89T^{2} \) |
| 97 | \( 1 - 9.16T + 97T^{2} \) |
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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
−10.73758418443309982185241429532, −9.571824031162981338148081032937, −9.045007811544518091860466757325, −8.020960220765039459603850781411, −7.53835184946990034689489298925, −5.85763788500352606391259150761, −5.06183371645896985350563842519, −4.11506870717107218672631645188, −2.98722859691152930322077668477, −1.33320863112958086616435196050,
1.33320863112958086616435196050, 2.98722859691152930322077668477, 4.11506870717107218672631645188, 5.06183371645896985350563842519, 5.85763788500352606391259150761, 7.53835184946990034689489298925, 8.020960220765039459603850781411, 9.045007811544518091860466757325, 9.571824031162981338148081032937, 10.73758418443309982185241429532