| L(s) = 1 | − 42.7·2-s + (124. − 216. i)3-s + 1.31e3·4-s + (1.21e3 − 2.10e3i)5-s + (−5.33e3 + 9.24e3i)6-s + (−2.67e3 − 4.63e3i)7-s − 3.44e4·8-s + (−2.13e4 − 3.69e4i)9-s + (−5.18e4 + 8.98e4i)10-s + 1.19e3·11-s + (1.64e5 − 2.84e5i)12-s + (−1.22e4 − 2.11e4i)13-s + (1.14e5 + 1.98e5i)14-s + (−3.02e5 − 5.24e5i)15-s + 7.99e5·16-s + (4.88e4 + 8.46e4i)17-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + (0.889 − 1.54i)3-s + 2.57·4-s + (0.868 − 1.50i)5-s + (−1.68 + 2.91i)6-s + (−0.421 − 0.730i)7-s − 2.97·8-s + (−1.08 − 1.87i)9-s + (−1.64 + 2.84i)10-s + 0.0245·11-s + (2.28 − 3.96i)12-s + (−0.118 − 0.205i)13-s + (0.797 + 1.38i)14-s + (−1.54 − 2.67i)15-s + 3.04·16-s + (0.141 + 0.245i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 - 0.459i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.887 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.263731 + 1.08264i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.263731 + 1.08264i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-2.16e7 + 5.85e6i)T \) |
| good | 2 | \( 1 + 42.7T + 512T^{2} \) |
| 3 | \( 1 + (-124. + 216. i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-1.21e3 + 2.10e3i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 7 | \( 1 + (2.67e3 + 4.63e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 - 1.19e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + (1.22e4 + 2.11e4i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 + (-4.88e4 - 8.46e4i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (1.40e5 - 2.43e5i)T + (-1.61e11 - 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-1.19e6 + 2.07e6i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (-3.04e6 - 5.27e6i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 + (-9.64e4 + 1.67e5i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (9.97e5 - 1.72e6i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 - 2.56e7T + 3.27e14T^{2} \) |
| 47 | \( 1 - 1.53e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + (2.92e6 - 5.07e6i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + 3.88e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + (1.34e7 + 2.33e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (1.54e8 - 2.67e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (-1.81e8 - 3.14e8i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 + (1.10e8 + 1.92e8i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (8.97e7 + 1.55e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-1.73e8 + 2.99e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (2.65e8 - 4.60e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 - 8.80e8T + 7.60e17T^{2} \) |
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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
−12.95015752141712172068529076100, −12.33707044469109179429962937555, −10.35973966484268743306391330359, −9.051605264449819254935921122969, −8.524619394783150847042204346010, −7.37152666630079902320769565838, −6.26293618022356318213202700516, −2.49939519416051920034804235899, −1.25878572929131754980979027825, −0.69180001884978053198150108953,
2.33799687318327605759011862090, 3.04485463400521341376460742645, 6.04383876935420051099117234178, 7.55522987754274358410765337580, 9.147537043069653385150497442881, 9.557395244815361364887803496990, 10.48249385448988881806185580618, 11.30330910288146117299905538593, 14.06286292639553894855735218055, 15.27758365875827518685645865321
