| L(s) = 1 | − 61.9i·2-s + (176. − 101. i)3-s − 2.81e3·4-s + (−2.89e3 + 1.67e3i)5-s + (−6.30e3 − 1.09e4i)6-s + (1.49e4 + 8.64e3i)7-s + 1.10e5i·8-s + (−8.78e3 + 1.52e4i)9-s + (1.03e5 + 1.79e5i)10-s − 7.93e4·11-s + (−4.95e5 + 2.86e5i)12-s + (3.52e5 − 6.09e5i)13-s + (5.35e5 − 9.26e5i)14-s + (−3.40e5 + 5.89e5i)15-s + 3.97e6·16-s + (−7.06e5 + 1.22e6i)17-s + ⋯ |
| L(s) = 1 | − 1.93i·2-s + (0.725 − 0.419i)3-s − 2.74·4-s + (−0.926 + 0.535i)5-s + (−0.811 − 1.40i)6-s + (0.890 + 0.514i)7-s + 3.37i·8-s + (−0.148 + 0.257i)9-s + (1.03 + 1.79i)10-s − 0.492·11-s + (−1.99 + 1.15i)12-s + (0.948 − 1.64i)13-s + (0.994 − 1.72i)14-s + (−0.448 + 0.776i)15-s + 3.79·16-s + (−0.497 + 0.861i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.25828 - 0.693169i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25828 - 0.693169i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.30e8 - 6.75e7i)T \) |
| good | 2 | \( 1 + 61.9iT - 1.02e3T^{2} \) |
| 3 | \( 1 + (-176. + 101. i)T + (2.95e4 - 5.11e4i)T^{2} \) |
| 5 | \( 1 + (2.89e3 - 1.67e3i)T + (4.88e6 - 8.45e6i)T^{2} \) |
| 7 | \( 1 + (-1.49e4 - 8.64e3i)T + (1.41e8 + 2.44e8i)T^{2} \) |
| 11 | \( 1 + 7.93e4T + 2.59e10T^{2} \) |
| 13 | \( 1 + (-3.52e5 + 6.09e5i)T + (-6.89e10 - 1.19e11i)T^{2} \) |
| 17 | \( 1 + (7.06e5 - 1.22e6i)T + (-1.00e12 - 1.74e12i)T^{2} \) |
| 19 | \( 1 + (-9.63e4 + 5.56e4i)T + (3.06e12 - 5.30e12i)T^{2} \) |
| 23 | \( 1 + (1.01e6 + 1.75e6i)T + (-2.07e13 + 3.58e13i)T^{2} \) |
| 29 | \( 1 + (-2.90e7 - 1.68e7i)T + (2.10e14 + 3.64e14i)T^{2} \) |
| 31 | \( 1 + (-2.36e7 - 4.09e7i)T + (-4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 + (-9.18e6 + 5.30e6i)T + (2.40e15 - 4.16e15i)T^{2} \) |
| 41 | \( 1 - 1.42e7T + 1.34e16T^{2} \) |
| 47 | \( 1 - 4.75e7T + 5.25e16T^{2} \) |
| 53 | \( 1 + (1.96e7 + 3.40e7i)T + (-8.74e16 + 1.51e17i)T^{2} \) |
| 59 | \( 1 + 9.82e8T + 5.11e17T^{2} \) |
| 61 | \( 1 + (-1.25e9 - 7.24e8i)T + (3.56e17 + 6.17e17i)T^{2} \) |
| 67 | \( 1 + (1.82e8 + 3.16e8i)T + (-9.11e17 + 1.57e18i)T^{2} \) |
| 71 | \( 1 + (-1.93e9 - 1.11e9i)T + (1.62e18 + 2.81e18i)T^{2} \) |
| 73 | \( 1 + (-8.39e8 - 4.84e8i)T + (2.14e18 + 3.72e18i)T^{2} \) |
| 79 | \( 1 + (-9.60e8 + 1.66e9i)T + (-4.73e18 - 8.19e18i)T^{2} \) |
| 83 | \( 1 + (-2.87e9 - 4.97e9i)T + (-7.75e18 + 1.34e19i)T^{2} \) |
| 89 | \( 1 + (4.48e9 - 2.58e9i)T + (1.55e19 - 2.70e19i)T^{2} \) |
| 97 | \( 1 + 6.89e9T + 7.37e19T^{2} \) |
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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
−13.22236137516928632983970299459, −12.27371460082010339315471635380, −11.05983794200916945088868351405, −10.50243870980068676548050800885, −8.357887468006322466423908735109, −8.270623307077937732372475796672, −5.05805639704752144258732080294, −3.44770180945814791299582836879, −2.58587840257146748529369640952, −1.20880524342932769730717094797,
0.51059271262711553398468056319, 4.04194913175046732541778935056, 4.64079018223996933887031122877, 6.48267192774054431020232692235, 7.87365798366460379923676168160, 8.491244599670670244081856635230, 9.514929531178643214461832390819, 11.70189853745850891758833281756, 13.64986681193704828235942473776, 14.11777579141135526172818348235
