| L(s) = 1 | + 1.71·2-s + 11.7·3-s − 29.0·4-s − 25·5-s + 20.1·6-s − 141.·7-s − 104.·8-s − 105.·9-s − 42.8·10-s − 121·11-s − 341.·12-s + 164.·13-s − 242.·14-s − 293.·15-s + 750.·16-s − 2.00e3·17-s − 180.·18-s + 361·19-s + 726.·20-s − 1.65e3·21-s − 207.·22-s − 2.67e3·23-s − 1.22e3·24-s + 625·25-s + 282.·26-s − 4.08e3·27-s + 4.10e3·28-s + ⋯ |
| L(s) = 1 | + 0.303·2-s + 0.753·3-s − 0.908·4-s − 0.447·5-s + 0.228·6-s − 1.08·7-s − 0.578·8-s − 0.432·9-s − 0.135·10-s − 0.301·11-s − 0.683·12-s + 0.270·13-s − 0.330·14-s − 0.336·15-s + 0.732·16-s − 1.68·17-s − 0.131·18-s + 0.229·19-s + 0.406·20-s − 0.820·21-s − 0.0914·22-s − 1.05·23-s − 0.435·24-s + 0.200·25-s + 0.0818·26-s − 1.07·27-s + 0.989·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1503407023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1503407023\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 - 1.71T + 32T^{2} \) |
| 3 | \( 1 - 11.7T + 243T^{2} \) |
| 7 | \( 1 + 141.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 164.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.00e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 2.67e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.62e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.02e4T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.11e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.94T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.57e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.75e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 9.33e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 56.2T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.85e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.10e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.23e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.30e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.02e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.25e5T + 8.58e9T^{2} \) |
| 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
−9.307123192677524136166390443557, −8.413637039427981477627821064437, −7.77433497652145879563298134601, −6.57286498456007037923846251341, −5.77331125576160472971349352011, −4.67831679230847416128522196503, −3.70484427951843944281257339692, −3.21647845279790291461150418057, −2.04006513116039256592540486435, −0.14419073388448932213991284196,
0.14419073388448932213991284196, 2.04006513116039256592540486435, 3.21647845279790291461150418057, 3.70484427951843944281257339692, 4.67831679230847416128522196503, 5.77331125576160472971349352011, 6.57286498456007037923846251341, 7.77433497652145879563298134601, 8.413637039427981477627821064437, 9.307123192677524136166390443557
