| L(s) = 1 | + (−2.40 + 2.66i)2-s + (−0.888 + 0.0933i)3-s + (−0.930 − 8.84i)4-s + (2.13 + 4.51i)5-s + (1.88 − 2.59i)6-s + (−0.0558 + 6.99i)7-s + (14.2 + 10.3i)8-s + (−8.02 + 1.70i)9-s + (−17.2 − 5.15i)10-s + (−1.37 − 0.291i)11-s + (1.65 + 7.77i)12-s + (−1.54 − 0.501i)13-s + (−18.5 − 16.9i)14-s + (−2.32 − 3.81i)15-s + (−26.9 + 5.73i)16-s + (−0.339 − 0.763i)17-s + ⋯ |
| L(s) = 1 | + (−1.20 + 1.33i)2-s + (−0.296 + 0.0311i)3-s + (−0.232 − 2.21i)4-s + (0.427 + 0.903i)5-s + (0.314 − 0.432i)6-s + (−0.00798 + 0.999i)7-s + (1.77 + 1.29i)8-s + (−0.891 + 0.189i)9-s + (−1.72 − 0.515i)10-s + (−0.124 − 0.0265i)11-s + (0.137 + 0.648i)12-s + (−0.118 − 0.0385i)13-s + (−1.32 − 1.21i)14-s + (−0.154 − 0.254i)15-s + (−1.68 + 0.358i)16-s + (−0.0199 − 0.0449i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.291 + 0.956i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.291 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.158863 - 0.214446i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.158863 - 0.214446i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.13 - 4.51i)T \) |
| 7 | \( 1 + (0.0558 - 6.99i)T \) |
| good | 2 | \( 1 + (2.40 - 2.66i)T + (-0.418 - 3.97i)T^{2} \) |
| 3 | \( 1 + (0.888 - 0.0933i)T + (8.80 - 1.87i)T^{2} \) |
| 11 | \( 1 + (1.37 + 0.291i)T + (110. + 49.2i)T^{2} \) |
| 13 | \( 1 + (1.54 + 0.501i)T + (136. + 99.3i)T^{2} \) |
| 17 | \( 1 + (0.339 + 0.763i)T + (-193. + 214. i)T^{2} \) |
| 19 | \( 1 + (28.9 + 3.04i)T + (353. + 75.0i)T^{2} \) |
| 23 | \( 1 + (-10.1 + 11.3i)T + (-55.2 - 526. i)T^{2} \) |
| 29 | \( 1 + (-41.5 + 30.2i)T + (259. - 799. i)T^{2} \) |
| 31 | \( 1 + (7.97 + 17.9i)T + (-643. + 714. i)T^{2} \) |
| 37 | \( 1 + (42.5 - 9.03i)T + (1.25e3 - 556. i)T^{2} \) |
| 41 | \( 1 + (53.0 + 17.2i)T + (1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + 44.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-1.06 + 2.39i)T + (-1.47e3 - 1.64e3i)T^{2} \) |
| 53 | \( 1 + (-5.69 - 54.1i)T + (-2.74e3 + 584. i)T^{2} \) |
| 59 | \( 1 + (20.6 - 18.5i)T + (363. - 3.46e3i)T^{2} \) |
| 61 | \( 1 + (-84.1 - 75.8i)T + (388. + 3.70e3i)T^{2} \) |
| 67 | \( 1 + (-37.8 + 16.8i)T + (3.00e3 - 3.33e3i)T^{2} \) |
| 71 | \( 1 + (36.7 - 26.7i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (7.89 - 37.1i)T + (-4.86e3 - 2.16e3i)T^{2} \) |
| 79 | \( 1 + (8.27 + 3.68i)T + (4.17e3 + 4.63e3i)T^{2} \) |
| 83 | \( 1 + (60.0 - 82.5i)T + (-2.12e3 - 6.55e3i)T^{2} \) |
| 89 | \( 1 + (-46.7 - 42.1i)T + (827. + 7.87e3i)T^{2} \) |
| 97 | \( 1 + (-93.0 - 128. i)T + (-2.90e3 + 8.94e3i)T^{2} \) |
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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
−13.55945396204723120519419016564, −11.91226438242408179993412979367, −10.76308942882544054613483432610, −10.03280497417219143336347344127, −8.823465071127544948586872149700, −8.240344438752763785420871763617, −6.79532074028311531175069211508, −6.15704683477207266701963947100, −5.21964228439094954051302001757, −2.43693691332073018137395717034,
0.23961320946143646557621717825, 1.67414528028699025466343688478, 3.38668660464847853491597520743, 4.90519031115390869582411839167, 6.73324519715014335215820714345, 8.302333070938523340986225056244, 8.780795453980504267925246517496, 10.00429743857564456495041465094, 10.62977721052690177291443081991, 11.60084879753997557774910116411
