| L(s) = 1 | + 4.36·2-s + (0.424 + 0.735i)3-s − 12.9·4-s + (−30.5 − 52.8i)5-s + (1.85 + 3.21i)6-s + (93.7 − 162. i)7-s − 196.·8-s + (121. − 209. i)9-s + (−133. − 230. i)10-s + 227.·11-s + (−5.49 − 9.52i)12-s + (−266. + 461. i)13-s + (409. − 708. i)14-s + (25.9 − 44.9i)15-s − 442.·16-s + (−497. + 862. i)17-s + ⋯ |
| L(s) = 1 | + 0.771·2-s + (0.0272 + 0.0472i)3-s − 0.404·4-s + (−0.546 − 0.946i)5-s + (0.0210 + 0.0364i)6-s + (0.722 − 1.25i)7-s − 1.08·8-s + (0.498 − 0.863i)9-s + (−0.421 − 0.730i)10-s + 0.567·11-s + (−0.0110 − 0.0190i)12-s + (−0.437 + 0.757i)13-s + (0.557 − 0.966i)14-s + (0.0297 − 0.0515i)15-s − 0.431·16-s + (−0.417 + 0.723i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 + 0.991i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.14431 - 1.29935i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14431 - 1.29935i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-9.25e3 + 7.82e3i)T \) |
| good | 2 | \( 1 - 4.36T + 32T^{2} \) |
| 3 | \( 1 + (-0.424 - 0.735i)T + (-121.5 + 210. i)T^{2} \) |
| 5 | \( 1 + (30.5 + 52.8i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-93.7 + 162. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 - 227.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (266. - 461. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + (497. - 862. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (1.04e3 + 1.80e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-574. - 994. i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-2.12e3 + 3.68e3i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-369. - 640. i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-8.28e3 - 1.43e4i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 - 8.48e3T + 1.15e8T^{2} \) |
| 47 | \( 1 - 8.27e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + (1.40e4 + 2.43e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 - 8.72e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + (-2.07e4 + 3.59e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (2.10e4 + 3.65e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (7.14e3 - 1.23e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (3.97e4 - 6.87e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (2.53e4 - 4.39e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-3.63e4 - 6.28e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-2.28e4 - 3.95e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + 1.04e5T + 8.58e9T^{2} \) |
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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
−14.50780696997224215632771950227, −13.43822831005731057042269495754, −12.47562506039789548506733130146, −11.38772051524214200297407869731, −9.565767919740217465277904377814, −8.402949163966807508476106553794, −6.70264336815725118601605178885, −4.55625445576868699920285325584, −4.11788797611165263798491228219, −0.794457863064576057927519898763,
2.65771831480667071451190016326, 4.44371344733715937826598773202, 5.80443288626557619079366234184, 7.61279726227992021432285892779, 8.975856576516300921623866924169, 10.69656980296535430023153795043, 11.92840212502485979633672990893, 12.88071554094855644251606327809, 14.44806274110838621851979994752, 14.79311039429299488845538903283
