L(s) = 1 | + (2.11 − 0.567i)2-s + (2.43 − 1.40i)4-s + (3.00 − 3.00i)5-s + (−2.60 − 0.481i)7-s + (1.26 − 1.26i)8-s + (4.65 − 8.06i)10-s + (0.698 + 2.60i)11-s + (0.373 − 3.58i)13-s + (−5.78 + 0.455i)14-s + (−0.857 + 1.48i)16-s + (−0.599 − 1.03i)17-s + (1.89 + 0.507i)19-s + (3.09 − 11.5i)20-s + (2.95 + 5.12i)22-s + (4.65 + 2.68i)23-s + ⋯ |
L(s) = 1 | + (1.49 − 0.401i)2-s + (1.21 − 0.703i)4-s + (1.34 − 1.34i)5-s + (−0.983 − 0.182i)7-s + (0.445 − 0.445i)8-s + (1.47 − 2.55i)10-s + (0.210 + 0.785i)11-s + (0.103 − 0.994i)13-s + (−1.54 + 0.121i)14-s + (−0.214 + 0.371i)16-s + (−0.145 − 0.251i)17-s + (0.434 + 0.116i)19-s + (0.691 − 2.57i)20-s + (0.630 + 1.09i)22-s + (0.970 + 0.560i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.259 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.06616 - 2.35149i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.06616 - 2.35149i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.60 + 0.481i)T \) |
| 13 | \( 1 + (-0.373 + 3.58i)T \) |
good | 2 | \( 1 + (-2.11 + 0.567i)T + (1.73 - i)T^{2} \) |
| 5 | \( 1 + (-3.00 + 3.00i)T - 5iT^{2} \) |
| 11 | \( 1 + (-0.698 - 2.60i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (0.599 + 1.03i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.89 - 0.507i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-4.65 - 2.68i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.47 - 2.56i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.36 - 3.36i)T - 31iT^{2} \) |
| 37 | \( 1 + (1.03 + 3.87i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-1.42 - 5.32i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (9.78 - 5.64i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.97 - 2.97i)T + 47iT^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 + (-3.14 + 11.7i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (4.55 - 2.63i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.15 + 1.11i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (0.800 - 2.98i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-4.78 - 4.78i)T + 73iT^{2} \) |
| 79 | \( 1 + 1.16T + 79T^{2} \) |
| 83 | \( 1 + (-3.24 + 3.24i)T - 83iT^{2} \) |
| 89 | \( 1 + (4.80 - 1.28i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (14.7 + 3.95i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.00581859077209774950981941348, −9.459509049691466369619614066110, −8.588721804441166734787364688766, −7.07698643382756098924351636690, −6.13182117927319971897145764947, −5.35357239377649177033166830565, −4.87204130774972665624560446112, −3.65887122282457717844733954746, −2.60311106421771857603736777723, −1.35379734487589164228753890301,
2.26318200108163526256549332782, 3.10970999440096201989146975787, 3.90873322373711871281185210110, 5.42091911842234235394933702339, 5.95222750410285988741150544166, 6.78144119122158585142079263577, 7.01734300979522847889759679993, 8.914271147540302068368334307495, 9.626743716363486318458675705476, 10.52843489610986978123397955004