| L(s) = 1 | + (0.305 − 1.97i)2-s + (−3.81 − 1.20i)4-s + 0.329·7-s + (−3.55 + 7.16i)8-s − 20.4i·11-s − 0.416i·13-s + (0.100 − 0.652i)14-s + (13.0 + 9.21i)16-s − 18.5i·17-s + 12.4i·19-s + (−40.5 − 6.26i)22-s + 23.2·23-s + (−0.823 − 0.127i)26-s + (−1.25 − 0.398i)28-s − 23.9·29-s + ⋯ |
| L(s) = 1 | + (0.152 − 0.988i)2-s + (−0.953 − 0.302i)4-s + 0.0471·7-s + (−0.444 + 0.895i)8-s − 1.86i·11-s − 0.0320i·13-s + (0.00720 − 0.0465i)14-s + (0.817 + 0.575i)16-s − 1.09i·17-s + 0.655i·19-s + (−1.84 − 0.284i)22-s + 1.01·23-s + (−0.0316 − 0.00489i)26-s + (−0.0449 − 0.0142i)28-s − 0.824·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7711824996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7711824996\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.305 + 1.97i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 0.329T + 49T^{2} \) |
| 11 | \( 1 + 20.4iT - 121T^{2} \) |
| 13 | \( 1 + 0.416iT - 169T^{2} \) |
| 17 | \( 1 + 18.5iT - 289T^{2} \) |
| 19 | \( 1 - 12.4iT - 361T^{2} \) |
| 23 | \( 1 - 23.2T + 529T^{2} \) |
| 29 | \( 1 + 23.9T + 841T^{2} \) |
| 31 | \( 1 - 42.0iT - 961T^{2} \) |
| 37 | \( 1 + 50.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 46.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 55.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 81.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 29.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 24.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 74.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 72.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 39.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 46.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 101. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 5.88T + 6.88e3T^{2} \) |
| 89 | \( 1 + 61.0T + 7.92e3T^{2} \) |
| 97 | \( 1 + 95.5iT - 9.40e3T^{2} \) |
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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
−9.447334360243540283744280772336, −8.729047942195851611580186666187, −8.025461977561960543179273644946, −6.69851003239052843068198938573, −5.58726796243068347679479652342, −4.93679013632297760863470601043, −3.53043807368218787628468455265, −3.02390080315863602117067476394, −1.50089882768097612901954102052, −0.24291067041490851694938003733,
1.74844549468098156533544457029, 3.37534177066317923117656750585, 4.54505350485937635689186543621, 5.06438062275629944861803196060, 6.34263288087366332906535548236, 6.96064590809437933179290380557, 7.79468767067499523963683633337, 8.578916639843810812351602283926, 9.619021603705464561702245356544, 10.01909235193082276174760144045
