| L(s) = 1 | + (−1.28 − 3.54i)2-s + (1.70 + 0.300i)3-s + (−7.81 + 6.55i)4-s + (−6.09 − 5.11i)5-s + (−1.13 − 6.42i)6-s + (−2.17 − 3.76i)7-s + (20.2 + 11.6i)8-s + (2.81 + 1.02i)9-s + (−10.2 + 28.1i)10-s + (4.81 − 8.34i)11-s + (−15.3 + 8.83i)12-s + (11.2 − 1.98i)13-s + (−10.5 + 12.5i)14-s + (−8.85 − 10.5i)15-s + (8.21 − 46.5i)16-s + (3.02 − 1.09i)17-s + ⋯ |
| L(s) = 1 | + (−0.644 − 1.77i)2-s + (0.568 + 0.100i)3-s + (−1.95 + 1.63i)4-s + (−1.21 − 1.02i)5-s + (−0.188 − 1.07i)6-s + (−0.310 − 0.538i)7-s + (2.53 + 1.46i)8-s + (0.313 + 0.114i)9-s + (−1.02 + 2.81i)10-s + (0.437 − 0.758i)11-s + (−1.27 + 0.736i)12-s + (0.867 − 0.152i)13-s + (−0.753 + 0.897i)14-s + (−0.590 − 0.703i)15-s + (0.513 − 2.91i)16-s + (0.177 − 0.0646i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.124i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0434142 + 0.696181i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0434142 + 0.696181i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.70 - 0.300i)T \) |
| 19 | \( 1 + (-4.60 + 18.4i)T \) |
| good | 2 | \( 1 + (1.28 + 3.54i)T + (-3.06 + 2.57i)T^{2} \) |
| 5 | \( 1 + (6.09 + 5.11i)T + (4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (2.17 + 3.76i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-4.81 + 8.34i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-11.2 + 1.98i)T + (158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (-3.02 + 1.09i)T + (221. - 185. i)T^{2} \) |
| 23 | \( 1 + (19.2 - 16.1i)T + (91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (-7.54 + 20.7i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-3.93 + 2.27i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 38.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-20.0 - 3.53i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (-49.0 - 41.1i)T + (321. + 1.82e3i)T^{2} \) |
| 47 | \( 1 + (5.05 + 1.83i)T + (1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (52.6 + 62.6i)T + (-487. + 2.76e3i)T^{2} \) |
| 59 | \( 1 + (-23.5 - 64.7i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-11.0 + 9.24i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-6.23 + 17.1i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (8.50 - 10.1i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (-7.50 + 42.5i)T + (-5.00e3 - 1.82e3i)T^{2} \) |
| 79 | \( 1 + (49.5 + 8.72i)T + (5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-75.4 - 130. i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-56.0 + 9.88i)T + (7.44e3 - 2.70e3i)T^{2} \) |
| 97 | \( 1 + (45.3 + 124. i)T + (-7.20e3 + 6.04e3i)T^{2} \) |
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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
−13.75573856691715192884193807024, −13.01676907036794561334363286799, −11.85099644949857624039609482225, −11.11101954390163539700825730195, −9.685601715389307252032129547612, −8.671744230559331302708132609844, −7.899332957567684240131301054814, −4.32014560510209039689420651610, −3.38513956567891301602121759084, −0.862495436157663223462323295397,
3.99983667600739820954346440605, 6.15189865322444717443786377178, 7.22352753677257791327834440600, 8.086741143791208725063131263696, 9.186961393156041249190125037121, 10.52380514314062114253609132332, 12.35215629753863978068612644154, 14.17443609189559516637041232470, 14.65794914165145882093570423562, 15.69741678591637810818391821888
