L(s) = 1 | + (−0.707 + 1.22i)2-s + (−0.999 − 1.73i)4-s + (2.51 + 4.32i)5-s + (5.12 + 2.95i)7-s + 2.82·8-s + (−7.07 + 0.0209i)10-s + (−3.29 − 1.90i)11-s + (19.2 − 11.1i)13-s + (−7.24 + 4.18i)14-s + (−2.00 + 3.46i)16-s + 1.20·17-s + 29.3·19-s + (4.97 − 8.67i)20-s + (4.66 − 2.69i)22-s + (−8.07 − 13.9i)23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.502 + 0.864i)5-s + (0.732 + 0.422i)7-s + 0.353·8-s + (−0.707 + 0.00209i)10-s + (−0.299 − 0.173i)11-s + (1.48 − 0.854i)13-s + (−0.517 + 0.298i)14-s + (−0.125 + 0.216i)16-s + 0.0708·17-s + 1.54·19-s + (0.248 − 0.433i)20-s + (0.212 − 0.122i)22-s + (−0.350 − 0.607i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.004153452\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.004153452\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 1.22i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.51 - 4.32i)T \) |
good | 7 | \( 1 + (-5.12 - 2.95i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (3.29 + 1.90i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-19.2 + 11.1i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 1.20T + 289T^{2} \) |
| 19 | \( 1 - 29.3T + 361T^{2} \) |
| 23 | \( 1 + (8.07 + 13.9i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-39.2 - 22.6i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (21.8 + 37.9i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 48.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (2.35 - 1.36i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-17.3 - 9.99i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (12.1 - 21.0i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 10.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-16.7 + 9.69i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (9.19 - 15.9i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-71.9 + 41.5i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 121. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 105. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (47.1 - 81.5i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-47.3 + 81.9i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 132. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-103. - 59.5i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.19125370271163592556770160208, −9.292686764944788766106284315088, −8.354510876398342987996687387412, −7.73635932255122215619467774207, −6.73565456138536230501466251114, −5.78908176207091798433396097658, −5.29907607264321970479654759249, −3.73490412521746520892448087762, −2.54820273266303509281673427037, −1.10627216659610562004462746410,
1.04600337402953229991885389696, 1.75108214040205425280748482218, 3.33818489420361967686348503115, 4.44315069144517455560970816227, 5.24131980939195608387284974050, 6.38221288706934735793754057271, 7.61423092839809224077862577276, 8.386276940126388127836967860519, 9.081705087502570242890725926225, 9.900346011539389053540101699152