| L(s) = 1 | + (2.08 − 5.25i)2-s − 10.4·3-s + (−23.2 − 21.9i)4-s + (−21.7 + 54.8i)6-s − 66.0i·7-s + (−164. + 76.6i)8-s − 134.·9-s + 141. i·11-s + (243. + 229. i)12-s − 246.·13-s + (−347. − 137. i)14-s + (60.5 + 1.02e3i)16-s − 297. i·17-s + (−279. + 705. i)18-s + 174. i·19-s + ⋯ |
| L(s) = 1 | + (0.368 − 0.929i)2-s − 0.669·3-s + (−0.727 − 0.685i)4-s + (−0.247 + 0.622i)6-s − 0.509i·7-s + (−0.905 + 0.423i)8-s − 0.551·9-s + 0.353i·11-s + (0.487 + 0.459i)12-s − 0.405·13-s + (−0.473 − 0.187i)14-s + (0.0591 + 0.998i)16-s − 0.249i·17-s + (−0.203 + 0.512i)18-s + 0.110i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0265i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9063029820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9063029820\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.08 + 5.25i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 10.4T + 243T^{2} \) |
| 7 | \( 1 + 66.0iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 141. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 246.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 297. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 174. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 1.57e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 922. iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 6.19e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.39e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.00e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.62e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.00e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.29e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.18e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 2.13e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 3.86e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.09e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.95e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 2.33e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.67e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.61e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.25e4iT - 8.58e9T^{2} \) |
| 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
−11.53041744135789330383282399489, −10.83360879039303292670518229344, −9.893736850494635893501961215199, −8.902228269127936494974019711180, −7.42913352730892353706299890175, −6.01523625563344786301204261602, −5.09108160364353890375637549474, −3.93517525520515850275865176900, −2.52383910721474801959054317241, −0.932786472175619675588098999572,
0.34579592353894659790119172834, 2.78532467817893638672953933141, 4.33484456320292542037778108029, 5.53498675910801375887300633696, 6.12163540784166160604635867305, 7.34031054822263067526607280466, 8.470613415009914892552677833370, 9.313973493751601358538692555154, 10.74488161452534659925438446653, 11.81596433472055299285186019097
