| L(s) = 1 | + (1.48 − 1.33i)2-s + (−2.82 + 0.757i)3-s + (0.421 − 3.97i)4-s + (−6.54 − 1.75i)5-s + (−3.18 + 4.90i)6-s + (−6.29 + 3.06i)7-s + (−4.69 − 6.47i)8-s + (−0.380 + 0.219i)9-s + (−12.0 + 6.14i)10-s + (18.1 − 4.87i)11-s + (1.82 + 11.5i)12-s + (−10.3 − 10.3i)13-s + (−5.26 + 12.9i)14-s + 19.8·15-s + (−15.6 − 3.34i)16-s + (−3.47 − 2.00i)17-s + ⋯ |
| L(s) = 1 | + (0.743 − 0.668i)2-s + (−0.942 + 0.252i)3-s + (0.105 − 0.994i)4-s + (−1.30 − 0.350i)5-s + (−0.531 + 0.817i)6-s + (−0.899 + 0.437i)7-s + (−0.586 − 0.809i)8-s + (−0.0422 + 0.0243i)9-s + (−1.20 + 0.614i)10-s + (1.65 − 0.443i)11-s + (0.151 + 0.963i)12-s + (−0.797 − 0.797i)13-s + (−0.375 + 0.926i)14-s + 1.32·15-s + (−0.977 − 0.209i)16-s + (−0.204 − 0.118i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0239i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00645260 + 0.539505i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00645260 + 0.539505i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.48 + 1.33i)T \) |
| 7 | \( 1 + (6.29 - 3.06i)T \) |
| good | 3 | \( 1 + (2.82 - 0.757i)T + (7.79 - 4.5i)T^{2} \) |
| 5 | \( 1 + (6.54 + 1.75i)T + (21.6 + 12.5i)T^{2} \) |
| 11 | \( 1 + (-18.1 + 4.87i)T + (104. - 60.5i)T^{2} \) |
| 13 | \( 1 + (10.3 + 10.3i)T + 169iT^{2} \) |
| 17 | \( 1 + (3.47 + 2.00i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-3.46 + 12.9i)T + (-312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (14.1 - 8.19i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (5.94 - 5.94i)T - 841iT^{2} \) |
| 31 | \( 1 + (-34.9 - 20.1i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-10.3 + 38.7i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + 37.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + (45.9 + 45.9i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (0.406 - 0.234i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (63.8 - 17.1i)T + (2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-19.0 - 71.1i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (3.16 - 11.8i)T + (-3.22e3 - 1.86e3i)T^{2} \) |
| 67 | \( 1 + (-16.2 - 60.7i)T + (-3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 - 20.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-57.2 + 99.1i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (46.0 + 79.7i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (10.7 + 10.7i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (17.3 + 30.1i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 67.9iT - 9.40e3T^{2} \) |
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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
−12.32111151075095173109475932575, −11.95945852716318524375501535686, −11.21055693902714900525560002299, −9.980865776580127395768366508537, −8.756241117137461939827782690809, −6.86296195412856148615113411225, −5.71462275433875653104602179085, −4.48969228678982283611394362139, −3.26986278200277437129446320295, −0.33832974208324294060597318096,
3.55785632797244020874300104774, 4.53793150513099735736926745903, 6.46613287083211430848613909297, 6.73354380727509905983870622643, 8.051608340618373778690134325549, 9.649887042080315722282938186513, 11.49150471448944787328443430186, 11.83482610080132172989295864436, 12.62132808294916795691574594661, 14.08767532111604143650254820189
