L(s) = 1 | + (−1.55 + 2.56i)3-s + 1.34i·7-s + (−4.17 − 7.97i)9-s + 3.66i·11-s − 7.34i·13-s − 6.31·17-s + 30.7·19-s + (−3.44 − 2.08i)21-s − 26.2·23-s + (26.9 + 1.68i)27-s − 40.9i·29-s + 9.97·31-s + (−9.40 − 5.69i)33-s + 39.4i·37-s + (18.8 + 11.4i)39-s + ⋯ |
L(s) = 1 | + (−0.517 + 0.855i)3-s + 0.191i·7-s + (−0.463 − 0.886i)9-s + 0.333i·11-s − 0.564i·13-s − 0.371·17-s + 1.61·19-s + (−0.164 − 0.0993i)21-s − 1.14·23-s + (0.998 + 0.0624i)27-s − 1.41i·29-s + 0.321·31-s + (−0.285 − 0.172i)33-s + 1.06i·37-s + (0.483 + 0.292i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.461442702\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.461442702\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.55 - 2.56i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 1.34iT - 49T^{2} \) |
| 11 | \( 1 - 3.66iT - 121T^{2} \) |
| 13 | \( 1 + 7.34iT - 169T^{2} \) |
| 17 | \( 1 + 6.31T + 289T^{2} \) |
| 19 | \( 1 - 30.7T + 361T^{2} \) |
| 23 | \( 1 + 26.2T + 529T^{2} \) |
| 29 | \( 1 + 40.9iT - 841T^{2} \) |
| 31 | \( 1 - 9.97T + 961T^{2} \) |
| 37 | \( 1 - 39.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 68.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 24.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 79.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 38.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 39.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 64.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 53.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 136. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 74.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 79.6T + 6.24e3T^{2} \) |
| 83 | \( 1 + 0.0506T + 6.88e3T^{2} \) |
| 89 | \( 1 + 22.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 35.0iT - 9.40e3T^{2} \) |
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
−9.806656731971219198818360606696, −8.994259579312703371856944445992, −8.051465374655642947063870196861, −7.13347872843501847183809659855, −6.00794036893663393992930495543, −5.45037250562837906072514013886, −4.46262383573342195572388703652, −3.61283135915296502706221127590, −2.46313391528711880785684225870, −0.71515314070457400625853468044,
0.77355512641255487776323242848, 1.90438523740037971923781564192, 3.12423516923346091851975451292, 4.38523081314736690364529759555, 5.44139090739704876772342888479, 6.12189594745794713711456682864, 7.11360266605656448450871829596, 7.60323558419089730735148222564, 8.607305077217569105291406922609, 9.413892721505063663617940728633