L(s) = 1 | − 2.69i·2-s − 5.27·4-s + (−0.933 − 0.539i)5-s + (2.30 + 1.29i)7-s + 8.84i·8-s + (−1.45 + 2.51i)10-s + (4.50 + 2.59i)11-s + (−1.40 + 3.31i)13-s + (3.49 − 6.22i)14-s + 13.3·16-s + 5.38·17-s + (−5.71 + 3.29i)19-s + (4.92 + 2.84i)20-s + (7.01 − 12.1i)22-s + 2.00·23-s + ⋯ |
L(s) = 1 | − 1.90i·2-s − 2.63·4-s + (−0.417 − 0.241i)5-s + (0.871 + 0.489i)7-s + 3.12i·8-s + (−0.459 + 0.796i)10-s + (1.35 + 0.783i)11-s + (−0.390 + 0.920i)13-s + (0.934 − 1.66i)14-s + 3.32·16-s + 1.30·17-s + (−1.31 + 0.756i)19-s + (1.10 + 0.636i)20-s + (1.49 − 2.58i)22-s + 0.417·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.183 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.183 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.00382 - 0.833690i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.00382 - 0.833690i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.30 - 1.29i)T \) |
| 13 | \( 1 + (1.40 - 3.31i)T \) |
good | 2 | \( 1 + 2.69iT - 2T^{2} \) |
| 5 | \( 1 + (0.933 + 0.539i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.50 - 2.59i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 - 5.38T + 17T^{2} \) |
| 19 | \( 1 + (5.71 - 3.29i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 2.00T + 23T^{2} \) |
| 29 | \( 1 + (-2.56 - 4.44i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.21 + 2.43i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 7.82iT - 37T^{2} \) |
| 41 | \( 1 + (-1.70 + 0.982i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.82 - 6.62i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (8.33 + 4.81i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.752 + 1.30i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 7.61iT - 59T^{2} \) |
| 61 | \( 1 + (2.74 + 4.74i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.83 - 2.79i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.55 - 4.93i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.88 + 2.82i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.65 + 2.85i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 8.99iT - 83T^{2} \) |
| 89 | \( 1 - 13.8iT - 89T^{2} \) |
| 97 | \( 1 + (-6.64 - 3.83i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.02486887047471204354400404674, −9.562347856061980978130227736286, −8.566196962567172285903175225800, −8.056073759510088721764179277473, −6.46352077578364035320755666791, −4.92902791255469055599818499282, −4.41905409073210360012153281852, −3.50041469398116802995190394578, −2.10506001789877401479312130051, −1.32933655950846650303917514023,
0.792733684806798454199719464124, 3.52293671022476257940970737374, 4.38602285796781314662800467686, 5.31347368342061732255327457450, 6.20110758403688447363433833683, 7.02554260045834976364212647296, 7.79249361674082604191689812046, 8.364197692574574388715668104905, 9.157241614213424172142680176881, 10.16823854319074003005162689575