| L(s) = 1 | + 1.59·3-s − 1.41·5-s − 0.109·7-s − 0.464·9-s + 3.16·11-s + 13-s − 2.25·15-s − 6.32·17-s + 6.49·19-s − 0.174·21-s − 8.02·23-s − 2.99·25-s − 5.51·27-s + 29-s + 6.62·31-s + 5.04·33-s + 0.155·35-s + 1.30·37-s + 1.59·39-s − 3.63·41-s − 10.8·43-s + 0.657·45-s + 6.28·47-s − 6.98·49-s − 10.0·51-s + 10.2·53-s − 4.48·55-s + ⋯ |
| L(s) = 1 | + 0.919·3-s − 0.633·5-s − 0.0415·7-s − 0.154·9-s + 0.954·11-s + 0.277·13-s − 0.582·15-s − 1.53·17-s + 1.48·19-s − 0.0381·21-s − 1.67·23-s − 0.599·25-s − 1.06·27-s + 0.185·29-s + 1.18·31-s + 0.877·33-s + 0.0262·35-s + 0.214·37-s + 0.254·39-s − 0.567·41-s − 1.65·43-s + 0.0980·45-s + 0.916·47-s − 0.998·49-s − 1.40·51-s + 1.41·53-s − 0.604·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 1.59T + 3T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 + 0.109T + 7T^{2} \) |
| 11 | \( 1 - 3.16T + 11T^{2} \) |
| 17 | \( 1 + 6.32T + 17T^{2} \) |
| 19 | \( 1 - 6.49T + 19T^{2} \) |
| 23 | \( 1 + 8.02T + 23T^{2} \) |
| 31 | \( 1 - 6.62T + 31T^{2} \) |
| 37 | \( 1 - 1.30T + 37T^{2} \) |
| 41 | \( 1 + 3.63T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 6.28T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 - 9.52T + 59T^{2} \) |
| 61 | \( 1 + 8.83T + 61T^{2} \) |
| 67 | \( 1 + 2.51T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 3.70T + 73T^{2} \) |
| 79 | \( 1 + 7.58T + 79T^{2} \) |
| 83 | \( 1 + 4.04T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.83007415009617695790413364965, −7.15836586154308992537111286466, −6.37620540491035363825742298881, −5.67903921313263594079768209385, −4.53162090001463695493123335382, −3.94579407044402219552722941112, −3.28842871132437333927134869348, −2.42411607273807744975796963617, −1.47404976925206405309265470462, 0,
1.47404976925206405309265470462, 2.42411607273807744975796963617, 3.28842871132437333927134869348, 3.94579407044402219552722941112, 4.53162090001463695493123335382, 5.67903921313263594079768209385, 6.37620540491035363825742298881, 7.15836586154308992537111286466, 7.83007415009617695790413364965
