| L(s) = 1 | + (0.938 − 4.41i)3-s + (−0.896 − 1.55i)5-s + (3.51 − 1.56i)7-s + (−10.3 − 4.62i)9-s + (−1.57 − 0.165i)11-s + (−10.8 − 9.72i)13-s + (−7.69 + 2.50i)15-s + (1.79 − 0.188i)17-s + (1.84 + 2.05i)19-s + (−3.60 − 16.9i)21-s + (20.8 + 28.6i)23-s + (10.8 − 18.8i)25-s + (−6.32 + 8.70i)27-s + (11.3 + 3.69i)29-s + (28.1 − 13.0i)31-s + ⋯ |
| L(s) = 1 | + (0.312 − 1.47i)3-s + (−0.179 − 0.310i)5-s + (0.501 − 0.223i)7-s + (−1.15 − 0.514i)9-s + (−0.143 − 0.0150i)11-s + (−0.831 − 0.748i)13-s + (−0.513 + 0.166i)15-s + (0.105 − 0.0110i)17-s + (0.0972 + 0.108i)19-s + (−0.171 − 0.808i)21-s + (0.906 + 1.24i)23-s + (0.435 − 0.754i)25-s + (−0.234 + 0.322i)27-s + (0.392 + 0.127i)29-s + (0.907 − 0.419i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.833186 - 1.18994i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.833186 - 1.18994i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + (-28.1 + 13.0i)T \) |
| good | 3 | \( 1 + (-0.938 + 4.41i)T + (-8.22 - 3.66i)T^{2} \) |
| 5 | \( 1 + (0.896 + 1.55i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (-3.51 + 1.56i)T + (32.7 - 36.4i)T^{2} \) |
| 11 | \( 1 + (1.57 + 0.165i)T + (118. + 25.1i)T^{2} \) |
| 13 | \( 1 + (10.8 + 9.72i)T + (17.6 + 168. i)T^{2} \) |
| 17 | \( 1 + (-1.79 + 0.188i)T + (282. - 60.0i)T^{2} \) |
| 19 | \( 1 + (-1.84 - 2.05i)T + (-37.7 + 359. i)T^{2} \) |
| 23 | \( 1 + (-20.8 - 28.6i)T + (-163. + 503. i)T^{2} \) |
| 29 | \( 1 + (-11.3 - 3.69i)T + (680. + 494. i)T^{2} \) |
| 37 | \( 1 + (-32.9 - 19.0i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-20.1 + 4.27i)T + (1.53e3 - 683. i)T^{2} \) |
| 43 | \( 1 + (24.6 - 22.1i)T + (193. - 1.83e3i)T^{2} \) |
| 47 | \( 1 + (-25.5 - 78.5i)T + (-1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-3.08 + 6.92i)T + (-1.87e3 - 2.08e3i)T^{2} \) |
| 59 | \( 1 + (71.7 + 15.2i)T + (3.18e3 + 1.41e3i)T^{2} \) |
| 61 | \( 1 + 110. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (-0.643 - 1.11i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-40.2 - 17.9i)T + (3.37e3 + 3.74e3i)T^{2} \) |
| 73 | \( 1 + (-11.6 - 1.22i)T + (5.21e3 + 1.10e3i)T^{2} \) |
| 79 | \( 1 + (-116. + 12.2i)T + (6.10e3 - 1.29e3i)T^{2} \) |
| 83 | \( 1 + (10.0 + 47.1i)T + (-6.29e3 + 2.80e3i)T^{2} \) |
| 89 | \( 1 + (62.8 - 86.5i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (94.9 + 68.9i)T + (2.90e3 + 8.94e3i)T^{2} \) |
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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
−12.81444238139531486893630096925, −12.14656141136672462836785492029, −11.02799892258037412320611776703, −9.550004905219507177148896056650, −8.078199414881214470589402174788, −7.65893368934416683684631232382, −6.39399718383904926843938480576, −4.88604489971790909658552255793, −2.77122667777234631900766326621, −1.08618127343304623741479120538,
2.78187900479904302281579991979, 4.28000089852530814439903896208, 5.16844545791947282190606838150, 6.96369058028940742882212753125, 8.445695176414490438037114411681, 9.351798693592037538040818388614, 10.34141233546123039490135197440, 11.16964575587799221316872790068, 12.26430786524338445950970252181, 13.80323443305648020137194059049
