| L(s) = 1 | + (−1.99 − 0.0598i)2-s + (−2.63 + 1.43i)3-s + (3.99 + 0.239i)4-s + (−1.71 − 9.75i)5-s + (5.35 − 2.70i)6-s + (4.26 + 3.57i)7-s + (−7.96 − 0.717i)8-s + (4.89 − 7.55i)9-s + (2.85 + 19.6i)10-s + (−2.25 + 12.8i)11-s + (−10.8 + 5.09i)12-s + (2.65 − 7.28i)13-s + (−8.30 − 7.40i)14-s + (18.5 + 23.2i)15-s + (15.8 + 1.91i)16-s + (−6.20 + 3.58i)17-s + ⋯ |
| L(s) = 1 | + (−0.999 − 0.0299i)2-s + (−0.878 + 0.477i)3-s + (0.998 + 0.0598i)4-s + (−0.343 − 1.95i)5-s + (0.892 − 0.451i)6-s + (0.608 + 0.510i)7-s + (−0.995 − 0.0897i)8-s + (0.543 − 0.839i)9-s + (0.285 + 1.96i)10-s + (−0.205 + 1.16i)11-s + (−0.905 + 0.424i)12-s + (0.204 − 0.560i)13-s + (−0.593 − 0.528i)14-s + (1.23 + 1.54i)15-s + (0.992 + 0.119i)16-s + (−0.365 + 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.124i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0117059 - 0.187712i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0117059 - 0.187712i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.99 + 0.0598i)T \) |
| 3 | \( 1 + (2.63 - 1.43i)T \) |
| good | 5 | \( 1 + (1.71 + 9.75i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-4.26 - 3.57i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (2.25 - 12.8i)T + (-113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-2.65 + 7.28i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (6.20 - 3.58i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (12.6 + 7.29i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (28.7 + 34.2i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-20.5 + 7.46i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (4.24 - 3.56i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (38.3 - 22.1i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (10.1 - 27.9i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (35.2 + 6.22i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (35.4 - 42.2i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + 4.60T + 2.80e3T^{2} \) |
| 59 | \( 1 + (1.03 + 5.84i)T + (-3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (15.9 - 18.9i)T + (-646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-17.6 + 48.5i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-4.25 + 2.45i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-12.4 + 21.6i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (23.6 - 8.62i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (48.7 - 17.7i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-67.3 - 38.9i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-16.0 + 90.8i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
| 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
−11.73482785303306521731098693848, −10.50394137498917471223466127242, −9.644640675648141290023654536693, −8.625155672475460725255663466462, −8.052945599157815702752564854212, −6.44404144261211516942237867177, −5.18042334828446556463478381392, −4.38825579329344689652188635655, −1.70088714072762380999333421820, −0.15571256290080077417856479542,
1.96106846338942476991256856951, 3.58804678399896482741474310617, 5.85161709080706269986405620436, 6.67927794112335351694837005431, 7.43515978754134310389439404245, 8.277815819836496610643927036404, 10.08080664574267505300542025899, 10.72789151250547600546888232538, 11.34252707999471676245120384725, 11.86671263993243730612617479110
