L(s) = 1 | + (−0.897 − 0.240i)2-s + (−0.920 − 1.46i)3-s + (−0.984 − 0.568i)4-s + (−2.48 − 0.667i)5-s + (0.473 + 1.53i)6-s + (−0.707 + 0.707i)7-s + (2.06 + 2.06i)8-s + (−1.30 + 2.70i)9-s + (2.07 + 1.19i)10-s + (0.573 − 2.14i)11-s + (0.0723 + 1.96i)12-s + (−1.08 + 3.43i)13-s + (0.804 − 0.464i)14-s + (1.31 + 4.26i)15-s + (−0.218 − 0.377i)16-s + (0.469 + 0.813i)17-s + ⋯ |
L(s) = 1 | + (−0.634 − 0.170i)2-s + (−0.531 − 0.847i)3-s + (−0.492 − 0.284i)4-s + (−1.11 − 0.298i)5-s + (0.193 + 0.628i)6-s + (−0.267 + 0.267i)7-s + (0.728 + 0.728i)8-s + (−0.435 + 0.900i)9-s + (0.656 + 0.378i)10-s + (0.173 − 0.645i)11-s + (0.0208 + 0.567i)12-s + (−0.300 + 0.953i)13-s + (0.215 − 0.124i)14-s + (0.339 + 1.10i)15-s + (−0.0545 − 0.0944i)16-s + (0.113 + 0.197i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.394699 - 0.131265i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.394699 - 0.131265i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.920 + 1.46i)T \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
| 13 | \( 1 + (1.08 - 3.43i)T \) |
good | 2 | \( 1 + (0.897 + 0.240i)T + (1.73 + i)T^{2} \) |
| 5 | \( 1 + (2.48 + 0.667i)T + (4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-0.573 + 2.14i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-0.469 - 0.813i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0928 - 0.346i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + 3.13T + 23T^{2} \) |
| 29 | \( 1 + (-2.72 + 1.57i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.12 - 7.93i)T + (-26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (1.50 + 5.60i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-2.37 + 2.37i)T - 41iT^{2} \) |
| 43 | \( 1 - 2.53iT - 43T^{2} \) |
| 47 | \( 1 + (-7.54 + 2.02i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + 13.1iT - 53T^{2} \) |
| 59 | \( 1 + (1.98 - 0.531i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 - 7.34T + 61T^{2} \) |
| 67 | \( 1 + (0.0555 + 0.0555i)T + 67iT^{2} \) |
| 71 | \( 1 + (4.59 + 1.23i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (4.23 - 4.23i)T - 73iT^{2} \) |
| 79 | \( 1 + (-1.26 + 2.19i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.48 - 9.25i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-12.5 + 3.36i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-12.1 - 12.1i)T + 97iT^{2} \) |
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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
−10.22845020867932278614557004205, −9.077193180784305281426831406544, −8.467820518024570945686856659790, −7.75726521438395770766131203154, −6.84467812946274677690268419760, −5.79132307535415245198688359724, −4.83538088207222893481154173624, −3.78557621653878885217498014391, −2.04393274457548457301141304530, −0.67043622726578636293027447837,
0.50949716997600040858491615016, 3.15515143007660275125641689584, 4.06052123610487758709837935645, 4.67700377349837916352710667265, 5.94863206216927778945448681753, 7.20857770291167972151344985727, 7.74276474575942066668721058275, 8.686244593683712601597785381579, 9.602845485350187239889601171209, 10.18247155498528607705188555747