| L(s) = 1 | − 1.65·2-s + 0.744·4-s − 1.18·5-s − 3.31·7-s + 2.07·8-s + 1.96·10-s + 2.10·11-s − 0.667·13-s + 5.49·14-s − 4.93·16-s + 5.75·17-s + 7.40·19-s − 0.884·20-s − 3.48·22-s + 23-s − 3.59·25-s + 1.10·26-s − 2.47·28-s + 29-s + 5.06·31-s + 4.01·32-s − 9.54·34-s + 3.93·35-s + 2.21·37-s − 12.2·38-s − 2.46·40-s − 0.348·41-s + ⋯ |
| L(s) = 1 | − 1.17·2-s + 0.372·4-s − 0.530·5-s − 1.25·7-s + 0.735·8-s + 0.621·10-s + 0.633·11-s − 0.185·13-s + 1.46·14-s − 1.23·16-s + 1.39·17-s + 1.69·19-s − 0.197·20-s − 0.742·22-s + 0.208·23-s − 0.718·25-s + 0.216·26-s − 0.466·28-s + 0.185·29-s + 0.909·31-s + 0.709·32-s − 1.63·34-s + 0.665·35-s + 0.363·37-s − 1.99·38-s − 0.390·40-s − 0.0545·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7385997774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7385997774\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.65T + 2T^{2} \) |
| 5 | \( 1 + 1.18T + 5T^{2} \) |
| 7 | \( 1 + 3.31T + 7T^{2} \) |
| 11 | \( 1 - 2.10T + 11T^{2} \) |
| 13 | \( 1 + 0.667T + 13T^{2} \) |
| 17 | \( 1 - 5.75T + 17T^{2} \) |
| 19 | \( 1 - 7.40T + 19T^{2} \) |
| 31 | \( 1 - 5.06T + 31T^{2} \) |
| 37 | \( 1 - 2.21T + 37T^{2} \) |
| 41 | \( 1 + 0.348T + 41T^{2} \) |
| 43 | \( 1 + 6.47T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 3.16T + 53T^{2} \) |
| 59 | \( 1 + 7.62T + 59T^{2} \) |
| 61 | \( 1 - 1.50T + 61T^{2} \) |
| 67 | \( 1 - 8.31T + 67T^{2} \) |
| 71 | \( 1 + 2.83T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 + 4.20T + 83T^{2} \) |
| 89 | \( 1 - 8.05T + 89T^{2} \) |
| 97 | \( 1 + 8.26T + 97T^{2} \) |
| 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
−8.066767762858840658880028263359, −7.48078657814727262679877066790, −7.00838809252610723369985359949, −6.11110222473136321438670790590, −5.32827365885027617211157944939, −4.29856511171562953607445197355, −3.50641620593705303960538670715, −2.82890091271565878707799140769, −1.39054987630488479297238685620, −0.59877653655617312867787432260,
0.59877653655617312867787432260, 1.39054987630488479297238685620, 2.82890091271565878707799140769, 3.50641620593705303960538670715, 4.29856511171562953607445197355, 5.32827365885027617211157944939, 6.11110222473136321438670790590, 7.00838809252610723369985359949, 7.48078657814727262679877066790, 8.066767762858840658880028263359
