| L(s) = 1 | − 2-s + 2·3-s − 5-s − 2·6-s + 8-s + 3·9-s + 10-s − 11-s − 2·15-s − 16-s − 17-s − 3·18-s + 22-s + 2·24-s + 4·27-s − 29-s + 2·30-s − 31-s − 2·33-s + 34-s − 40-s − 3·45-s − 2·48-s + 49-s − 2·51-s − 4·54-s + 55-s + ⋯ |
| L(s) = 1 | − 2-s + 2·3-s − 5-s − 2·6-s + 8-s + 3·9-s + 10-s − 11-s − 2·15-s − 16-s − 17-s − 3·18-s + 22-s + 2·24-s + 4·27-s − 29-s + 2·30-s − 31-s − 2·33-s + 34-s − 40-s − 3·45-s − 2·48-s + 49-s − 2·51-s − 4·54-s + 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6596961330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6596961330\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 239 | \( 1 - T \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 3 | \( ( 1 - T )^{2} \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−12.69608155627197418587696892777, −11.02216619875814071752637615084, −10.09143731745367419904081209986, −9.140749285674249756223589207224, −8.544081640314261424161672634927, −7.69989522885676126640744837562, −7.27397792517399702010481856985, −4.57630416073136307777408201020, −3.60015862115289006588450814931, −2.12721823689176445206298456852,
2.12721823689176445206298456852, 3.60015862115289006588450814931, 4.57630416073136307777408201020, 7.27397792517399702010481856985, 7.69989522885676126640744837562, 8.544081640314261424161672634927, 9.140749285674249756223589207224, 10.09143731745367419904081209986, 11.02216619875814071752637615084, 12.69608155627197418587696892777
