| 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 | \( 1 + 35.0iT - 9.40e3T^{2} \) |
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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
−9.413892721505063663617940728633, −8.607305077217569105291406922609, −7.60323558419089730735148222564, −7.11360266605656448450871829596, −6.12189594745794713711456682864, −5.44139090739704876772342888479, −4.38523081314736690364529759555, −3.12423516923346091851975451292, −1.90438523740037971923781564192, −0.77355512641255487776323242848,
0.71515314070457400625853468044, 2.46313391528711880785684225870, 3.61283135915296502706221127590, 4.46262383573342195572388703652, 5.45037250562837906072514013886, 6.00794036893663393992930495543, 7.13347872843501847183809659855, 8.051465374655642947063870196861, 8.994259579312703371856944445992, 9.806656731971219198818360606696
