| L(s) = 1 | − 55.4i·2-s + (−88.6 + 51.1i)3-s − 2.05e3·4-s + (−1.27e3 + 736. i)5-s + (2.83e3 + 4.91e3i)6-s + (−1.43e4 − 8.26e3i)7-s + 5.70e4i·8-s + (−2.42e4 + 4.20e4i)9-s + (4.08e4 + 7.07e4i)10-s + 2.40e4·11-s + (1.81e5 − 1.05e5i)12-s + (−1.64e5 + 2.84e5i)13-s + (−4.58e5 + 7.93e5i)14-s + (7.53e4 − 1.30e5i)15-s + 1.06e6·16-s + (1.06e6 − 1.84e6i)17-s + ⋯ |
| L(s) = 1 | − 1.73i·2-s + (−0.364 + 0.210i)3-s − 2.00·4-s + (−0.408 + 0.235i)5-s + (0.365 + 0.632i)6-s + (−0.851 − 0.491i)7-s + 1.74i·8-s + (−0.411 + 0.712i)9-s + (0.408 + 0.707i)10-s + 0.149·11-s + (0.731 − 0.422i)12-s + (−0.442 + 0.766i)13-s + (−0.851 + 1.47i)14-s + (0.0992 − 0.171i)15-s + 1.01·16-s + (0.751 − 1.30i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.360 + 0.932i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.360 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.685237 - 0.469754i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.685237 - 0.469754i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.41e8 - 4.10e7i)T \) |
| good | 2 | \( 1 + 55.4iT - 1.02e3T^{2} \) |
| 3 | \( 1 + (88.6 - 51.1i)T + (2.95e4 - 5.11e4i)T^{2} \) |
| 5 | \( 1 + (1.27e3 - 736. i)T + (4.88e6 - 8.45e6i)T^{2} \) |
| 7 | \( 1 + (1.43e4 + 8.26e3i)T + (1.41e8 + 2.44e8i)T^{2} \) |
| 11 | \( 1 - 2.40e4T + 2.59e10T^{2} \) |
| 13 | \( 1 + (1.64e5 - 2.84e5i)T + (-6.89e10 - 1.19e11i)T^{2} \) |
| 17 | \( 1 + (-1.06e6 + 1.84e6i)T + (-1.00e12 - 1.74e12i)T^{2} \) |
| 19 | \( 1 + (-2.62e6 + 1.51e6i)T + (3.06e12 - 5.30e12i)T^{2} \) |
| 23 | \( 1 + (-2.12e5 - 3.67e5i)T + (-2.07e13 + 3.58e13i)T^{2} \) |
| 29 | \( 1 + (1.41e7 + 8.17e6i)T + (2.10e14 + 3.64e14i)T^{2} \) |
| 31 | \( 1 + (-4.52e6 - 7.84e6i)T + (-4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 + (-1.34e6 + 7.75e5i)T + (2.40e15 - 4.16e15i)T^{2} \) |
| 41 | \( 1 - 5.83e7T + 1.34e16T^{2} \) |
| 47 | \( 1 - 3.25e8T + 5.25e16T^{2} \) |
| 53 | \( 1 + (-1.60e8 - 2.77e8i)T + (-8.74e16 + 1.51e17i)T^{2} \) |
| 59 | \( 1 - 1.07e9T + 5.11e17T^{2} \) |
| 61 | \( 1 + (-1.38e8 - 8.00e7i)T + (3.56e17 + 6.17e17i)T^{2} \) |
| 67 | \( 1 + (-2.36e8 - 4.09e8i)T + (-9.11e17 + 1.57e18i)T^{2} \) |
| 71 | \( 1 + (-7.23e8 - 4.17e8i)T + (1.62e18 + 2.81e18i)T^{2} \) |
| 73 | \( 1 + (7.56e8 + 4.36e8i)T + (2.14e18 + 3.72e18i)T^{2} \) |
| 79 | \( 1 + (7.34e8 - 1.27e9i)T + (-4.73e18 - 8.19e18i)T^{2} \) |
| 83 | \( 1 + (-3.14e9 - 5.45e9i)T + (-7.75e18 + 1.34e19i)T^{2} \) |
| 89 | \( 1 + (-1.61e8 + 9.32e7i)T + (1.55e19 - 2.70e19i)T^{2} \) |
| 97 | \( 1 - 1.35e10T + 7.37e19T^{2} \) |
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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
−13.30230741921519187466382566504, −11.86574834815933723343247769803, −11.34274614572884872907919181386, −10.10022051431880855462222217730, −9.288031209498779322470136954164, −7.31244312610456026764018590267, −5.10772902776249576323167697699, −3.71274773318769056648603492497, −2.56076941396767277884421564775, −0.73901236241898545189568333382,
0.46394902342388792332856444313, 3.61825762242132795919569170338, 5.49116002080483372861757133559, 6.20290941960914124988711102991, 7.52933048760335896930253389368, 8.635345396778219785041426866671, 9.863208703453074805847072865939, 12.03119877595851410999716598080, 12.92882568291981612147999165546, 14.43799220062264502535191026616
