L(s) = 1 | + 0.747i·2-s + 7.44·4-s + (−4.35 − 7.53i)5-s + (−11.6 + 14.4i)7-s + 11.5i·8-s + (5.63 − 3.25i)10-s + (35.9 + 20.7i)11-s + (33.0 + 19.0i)13-s + (−10.7 − 8.68i)14-s + 50.8·16-s + (39.9 + 69.2i)17-s + (−49.4 − 28.5i)19-s + (−32.3 − 56.0i)20-s + (−15.5 + 26.8i)22-s + (153. − 88.5i)23-s + ⋯ |
L(s) = 1 | + 0.264i·2-s + 0.930·4-s + (−0.389 − 0.674i)5-s + (−0.627 + 0.778i)7-s + 0.510i·8-s + (0.178 − 0.102i)10-s + (0.985 + 0.568i)11-s + (0.704 + 0.406i)13-s + (−0.205 − 0.165i)14-s + 0.795·16-s + (0.570 + 0.987i)17-s + (−0.596 − 0.344i)19-s + (−0.361 − 0.626i)20-s + (−0.150 + 0.260i)22-s + (1.38 − 0.802i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.86084 + 0.811779i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.86084 + 0.811779i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (11.6 - 14.4i)T \) |
good | 2 | \( 1 - 0.747iT - 8T^{2} \) |
| 5 | \( 1 + (4.35 + 7.53i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-35.9 - 20.7i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-33.0 - 19.0i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-39.9 - 69.2i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (49.4 + 28.5i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-153. + 88.5i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-60.1 + 34.7i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 260. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (74.9 - 129. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (54.7 - 94.7i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-124. - 215. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + 295.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-263. + 152. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + 720.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 648. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 194.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 98.7iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-226. + 130. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + (285. + 494. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-89.5 + 155. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-956. + 552. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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
−12.31805778555311909315823011876, −11.42364549375028554335086154858, −10.33161371623788552276976450228, −8.980747200875898831076422470696, −8.325151459360393462086588055884, −6.79926495423399781472683291546, −6.21737132649188902396095090448, −4.72090926641037491834798329763, −3.17229868392784134388909339126, −1.50980072619075947817261592919,
1.02442286875705981399440844373, 3.02911965056962965167033667342, 3.77895763898094680248402841187, 5.89010739876244493499069358008, 6.89223600776968996566266907681, 7.54908888406010909380084550342, 9.119187232319703690286090105563, 10.29456162284532627912045431657, 11.07197896529460826189615530727, 11.70018858813526593916911926790