L(s) = 1 | + (2.51 + 0.817i)2-s + (2.47 + 1.70i)3-s + (2.42 + 1.76i)4-s + (2.68 − 0.874i)5-s + (4.82 + 6.29i)6-s + (6.05 + 4.39i)7-s + (−1.55 − 2.14i)8-s + (3.21 + 8.40i)9-s + 7.48·10-s + (2.99 + 8.48i)12-s + (6.93 − 21.3i)13-s + (11.6 + 16.0i)14-s + (8.13 + 2.41i)15-s + (−5.87 − 18.0i)16-s + (−20.1 + 6.54i)17-s + (1.22 + 23.7i)18-s + ⋯ |
L(s) = 1 | + (1.25 + 0.408i)2-s + (0.823 + 0.566i)3-s + (0.606 + 0.440i)4-s + (0.537 − 0.174i)5-s + (0.804 + 1.04i)6-s + (0.864 + 0.628i)7-s + (−0.194 − 0.267i)8-s + (0.357 + 0.933i)9-s + 0.748·10-s + (0.249 + 0.707i)12-s + (0.533 − 1.64i)13-s + (0.831 + 1.14i)14-s + (0.542 + 0.160i)15-s + (−0.366 − 1.12i)16-s + (−1.18 + 0.384i)17-s + (0.0679 + 1.32i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(4.03888 + 2.06731i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.03888 + 2.06731i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.47 - 1.70i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-2.51 - 0.817i)T + (3.23 + 2.35i)T^{2} \) |
| 5 | \( 1 + (-2.68 + 0.874i)T + (20.2 - 14.6i)T^{2} \) |
| 7 | \( 1 + (-6.05 - 4.39i)T + (15.1 + 46.6i)T^{2} \) |
| 13 | \( 1 + (-6.93 + 21.3i)T + (-136. - 99.3i)T^{2} \) |
| 17 | \( 1 + (20.1 - 6.54i)T + (233. - 169. i)T^{2} \) |
| 19 | \( 1 + (12.1 - 8.79i)T + (111. - 343. i)T^{2} \) |
| 23 | \( 1 - 31.1iT - 529T^{2} \) |
| 29 | \( 1 + (-12.4 + 17.1i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (-9.27 + 28.5i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (-8.09 - 5.87i)T + (423. + 1.30e3i)T^{2} \) |
| 41 | \( 1 + (24.8 + 34.2i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + 14.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-21.6 - 29.7i)T + (-682. + 2.10e3i)T^{2} \) |
| 53 | \( 1 + (-40.3 - 13.1i)T + (2.27e3 + 1.65e3i)T^{2} \) |
| 59 | \( 1 + (19.9 - 27.4i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (30.0 + 92.5i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + 42T + 4.48e3T^{2} \) |
| 71 | \( 1 + (-61.8 + 20.1i)T + (4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (60.5 + 43.9i)T + (1.64e3 + 5.06e3i)T^{2} \) |
| 79 | \( 1 + (-6.93 + 21.3i)T + (-5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (20.1 - 6.54i)T + (5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 - 62.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-22.8 + 70.3i)T + (-7.61e3 - 5.53e3i)T^{2} \) |
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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
−11.46431447942066937145159253945, −10.43077720769475254436189035171, −9.434295772523017592248168734172, −8.482036044772868706395091304254, −7.65930615134357095421036031328, −6.04691777106440695856080604311, −5.39676533834773716837473997294, −4.42797278959457008314847655654, −3.37593004951499189215217776222, −2.08026163764716885518245776406,
1.73020923672717207748334134390, 2.61853274550684360087267324609, 4.11149425286637410266115948076, 4.66362792585603776365367932130, 6.35832588736785292694000384114, 6.90994220016756166099898599808, 8.455006587605443064714721137695, 8.966775739336093187279470512113, 10.44159877021263659872160414902, 11.37676380815928272480897003904