| L(s) = 1 | + (−1.30 + 0.541i)2-s + (0.420 + 2.78i)3-s + (1.41 − 1.41i)4-s + (−3.23 − 1.26i)5-s + (−2.06 − 3.41i)6-s + (0.157 + 2.64i)7-s + (−1.07 + 2.61i)8-s + (−4.73 + 1.46i)9-s + (4.90 − 0.0934i)10-s + (1.48 − 4.80i)11-s + (4.54 + 3.34i)12-s + (−5.09 + 1.16i)13-s + (−1.63 − 3.36i)14-s + (2.17 − 9.54i)15-s + (−0.00547 − 3.99i)16-s + (1.52 − 1.04i)17-s + ⋯ |
| L(s) = 1 | + (−0.923 + 0.382i)2-s + (0.242 + 1.61i)3-s + (0.706 − 0.707i)4-s + (−1.44 − 0.567i)5-s + (−0.841 − 1.39i)6-s + (0.0596 + 0.998i)7-s + (−0.381 + 0.924i)8-s + (−1.57 + 0.487i)9-s + (1.55 − 0.0295i)10-s + (0.447 − 1.45i)11-s + (1.31 + 0.966i)12-s + (−1.41 + 0.322i)13-s + (−0.437 − 0.899i)14-s + (0.562 − 2.46i)15-s + (−0.00136 − 0.999i)16-s + (0.370 − 0.252i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0551583 - 0.0695299i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0551583 - 0.0695299i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.30 - 0.541i)T \) |
| 7 | \( 1 + (-0.157 - 2.64i)T \) |
| good | 3 | \( 1 + (-0.420 - 2.78i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (3.23 + 1.26i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (-1.48 + 4.80i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (5.09 - 1.16i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-1.52 + 1.04i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (3.89 - 2.24i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.05 + 2.08i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (-1.89 + 3.94i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (-2.30 + 3.98i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.27 - 0.395i)T + (36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (0.683 + 0.857i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (-5.18 - 4.13i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (3.45 - 3.20i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (6.54 + 0.490i)T + (52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (7.97 - 3.12i)T + (43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (-5.57 + 0.417i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (-0.393 - 0.227i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.26 - 0.607i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (3.99 + 3.70i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (-1.16 - 2.02i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.89 + 1.57i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (15.3 - 4.74i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 + 0.457T + 97T^{2} \) |
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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
−11.67686164042253663598419857145, −10.93239921409049007661078599409, −9.886627294341824792783601556047, −9.128395382537719404935772929308, −8.429779201296552969763949458917, −7.87569059801191347322187055098, −6.18551857272823062891618178102, −5.08974561703579862403479752866, −4.12824608653519487993977315224, −2.82171458441098383190558955406,
0.07377888137642398033773619711, 1.72321441848496158655282622485, 3.02703010575288807915951122475, 4.32081119499875279665152408990, 6.85351952578593356477076162847, 7.06211909596708086067055240327, 7.70126807168262800045152955681, 8.409484284845683874801749788789, 9.841482123308326050513812744559, 10.71935862018183244027794465790
