| L(s) = 1 | + (−1 + 1.73i)2-s + (−1.99 − 3.46i)4-s + 7.99·8-s + (−10.8 + 18.7i)11-s + (−8 + 13.8i)16-s − 28.3·17-s + 2.30·19-s + (−21.6 − 37.5i)22-s + (−12.5 + 21.6i)25-s + (−15.9 − 27.7i)32-s + (28.3 − 49.1i)34-s + (−2.30 + 3.98i)38-s + (−17.8 − 30.9i)41-s + (−33.2 + 57.5i)43-s + 86.7·44-s + ⋯ |
| L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 0.999·8-s + (−0.986 + 1.70i)11-s + (−0.5 + 0.866i)16-s − 1.67·17-s + 0.121·19-s + (−0.986 − 1.70i)22-s + (−0.5 + 0.866i)25-s + (−0.499 − 0.866i)32-s + (0.835 − 1.44i)34-s + (−0.0606 + 0.104i)38-s + (−0.436 − 0.755i)41-s + (−0.773 + 1.33i)43-s + 1.97·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00922i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00236148 - 0.511753i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00236148 - 0.511753i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1 - 1.73i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (10.8 - 18.7i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 28.3T + 289T^{2} \) |
| 19 | \( 1 - 2.30T + 361T^{2} \) |
| 23 | \( 1 + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 + (17.8 + 30.9i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (33.2 - 57.5i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 + (-57.2 - 99.1i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-66.9 - 115. i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 100.T + 5.32e3T^{2} \) |
| 79 | \( 1 + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-79 + 136. i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 146T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-49.9 + 86.5i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
| 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
−12.88563898773963185201717588462, −11.43268032445095098840436086935, −10.32917009891751667845099986959, −9.593321844051798933997193141050, −8.539279138948929880449458575976, −7.45278819079553199638414103337, −6.73608954010980350475924301056, −5.32413880187821766657740871065, −4.39196292947973552380071960533, −2.05168015986657823888316682441,
0.31818183640242626108762187906, 2.35013170716717550082801988236, 3.59972957805768996764003003245, 5.02894901456723666273411390729, 6.52412037713734530892687853162, 8.043960930087855465230227610986, 8.609023505265173200641283558035, 9.745443013725003425952526262998, 10.87273313118746369693618673101, 11.26324197408941110681865937557
