| L(s) = 1 | − 0.394·2-s + 1.27·3-s − 1.84·4-s + 1.92·5-s − 0.502·6-s + 3.31·7-s + 1.51·8-s − 1.37·9-s − 0.760·10-s + 2.52·11-s − 2.34·12-s − 4.15·13-s − 1.30·14-s + 2.45·15-s + 3.09·16-s + 2.64·17-s + 0.543·18-s + 6.63·19-s − 3.55·20-s + 4.22·21-s − 0.996·22-s − 4.34·23-s + 1.93·24-s − 1.28·25-s + 1.64·26-s − 5.57·27-s − 6.11·28-s + ⋯ |
| L(s) = 1 | − 0.278·2-s + 0.735·3-s − 0.922·4-s + 0.862·5-s − 0.205·6-s + 1.25·7-s + 0.536·8-s − 0.459·9-s − 0.240·10-s + 0.761·11-s − 0.677·12-s − 1.15·13-s − 0.349·14-s + 0.634·15-s + 0.772·16-s + 0.640·17-s + 0.128·18-s + 1.52·19-s − 0.795·20-s + 0.921·21-s − 0.212·22-s − 0.905·23-s + 0.394·24-s − 0.256·25-s + 0.321·26-s − 1.07·27-s − 1.15·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.730369509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.730369509\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 619 | \( 1 - T \) |
| good | 2 | \( 1 + 0.394T + 2T^{2} \) |
| 3 | \( 1 - 1.27T + 3T^{2} \) |
| 5 | \( 1 - 1.92T + 5T^{2} \) |
| 7 | \( 1 - 3.31T + 7T^{2} \) |
| 11 | \( 1 - 2.52T + 11T^{2} \) |
| 13 | \( 1 + 4.15T + 13T^{2} \) |
| 17 | \( 1 - 2.64T + 17T^{2} \) |
| 19 | \( 1 - 6.63T + 19T^{2} \) |
| 23 | \( 1 + 4.34T + 23T^{2} \) |
| 29 | \( 1 - 0.725T + 29T^{2} \) |
| 31 | \( 1 - 6.55T + 31T^{2} \) |
| 37 | \( 1 - 6.31T + 37T^{2} \) |
| 41 | \( 1 + 0.0915T + 41T^{2} \) |
| 43 | \( 1 - 0.725T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 4.81T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 3.71T + 61T^{2} \) |
| 67 | \( 1 + 7.85T + 67T^{2} \) |
| 71 | \( 1 + 8.16T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 + 2.55T + 83T^{2} \) |
| 89 | \( 1 + 9.14T + 89T^{2} \) |
| 97 | \( 1 - 0.730T + 97T^{2} \) |
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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
−10.19847089956165860178441700948, −9.625490723393769058257237639125, −8.985020101052622270627111803999, −8.032317973588889519425251752808, −7.54791709402811663542247085130, −5.85678353986373043863107064432, −5.09592096654851356374539169410, −4.06676241454453968017860622484, −2.64436977122787549211386920970, −1.34785198323514141736890921103,
1.34785198323514141736890921103, 2.64436977122787549211386920970, 4.06676241454453968017860622484, 5.09592096654851356374539169410, 5.85678353986373043863107064432, 7.54791709402811663542247085130, 8.032317973588889519425251752808, 8.985020101052622270627111803999, 9.625490723393769058257237639125, 10.19847089956165860178441700948
