| L(s) = 1 | + (−2.55 − 1.57i)3-s + 9.08i·5-s + (3.71 − 6.43i)7-s + (4.03 + 8.04i)9-s + (−3.75 − 2.17i)11-s + (5.81 − 10.0i)13-s + (14.3 − 23.1i)15-s + (18.6 + 10.7i)17-s + (−9.10 − 16.6i)19-s + (−19.6 + 10.5i)21-s + (−3.34 − 1.93i)23-s − 57.4·25-s + (2.39 − 26.8i)27-s + 50.7i·29-s + (17.5 + 30.4i)31-s + ⋯ |
| L(s) = 1 | + (−0.850 − 0.525i)3-s + 1.81i·5-s + (0.531 − 0.919i)7-s + (0.447 + 0.894i)9-s + (−0.341 − 0.197i)11-s + (0.447 − 0.774i)13-s + (0.954 − 1.54i)15-s + (1.09 + 0.633i)17-s + (−0.478 − 0.877i)19-s + (−0.935 + 0.503i)21-s + (−0.145 − 0.0840i)23-s − 2.29·25-s + (0.0885 − 0.996i)27-s + 1.74i·29-s + (0.566 + 0.981i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.307236963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.307236963\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.55 + 1.57i)T \) |
| 19 | \( 1 + (9.10 + 16.6i)T \) |
| good | 5 | \( 1 - 9.08iT - 25T^{2} \) |
| 7 | \( 1 + (-3.71 + 6.43i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (3.75 + 2.17i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-5.81 + 10.0i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (-18.6 - 10.7i)T + (144.5 + 250. i)T^{2} \) |
| 23 | \( 1 + (3.34 + 1.93i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 50.7iT - 841T^{2} \) |
| 31 | \( 1 + (-17.5 - 30.4i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 22.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 39.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-3.94 - 6.82i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + 66.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-10.7 + 6.21i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 - 113. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 100.T + 3.72e3T^{2} \) |
| 67 | \( 1 + (36.7 - 63.6i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (2.86 + 1.65i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-17.9 + 31.0i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-63.5 - 110. i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (76.8 + 44.3i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-78.4 + 45.2i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-29.8 - 51.6i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.43504017047294831869297236279, −10.27625510762922582945229983884, −8.346469434692542059859217014931, −7.50200695039022549173729730276, −6.93089764170146216100493809968, −6.13501061818470173433025622966, −5.14687238877101914264624819911, −3.75531099793796060640050215957, −2.68396894702596270846976040409, −1.12749143232255632528793099018,
0.63313461964086959281373078443, 1.92347244353033243383546645013, 3.99997172343373491201226689871, 4.72277055123559457854853878324, 5.54804581062055770707502172948, 6.09483338729162465567385646389, 7.79804803342116921677479574661, 8.479258977634581537826157880964, 9.451132565916029815542724656440, 9.842115820293315317481288874410
