| L(s) = 1 | + (−9.77 + 12.2i)2-s + (−129. + 19.5i)3-s + (59.2 + 259. i)4-s + (828. − 565. i)5-s + (1.03e3 − 1.78e3i)6-s + (3.21e3 + 5.56e3i)7-s + (−1.09e4 − 5.29e3i)8-s + (−2.30e3 + 710. i)9-s + (−1.17e3 + 1.56e4i)10-s + (1.96e4 − 8.62e4i)11-s + (−1.27e4 − 3.25e4i)12-s + (−1.03e4 − 1.38e5i)13-s + (−9.95e4 − 1.50e4i)14-s + (−9.66e4 + 8.96e4i)15-s + (4.94e4 − 2.38e4i)16-s + (4.33e5 + 2.95e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.431 + 0.541i)2-s + (−0.926 + 0.139i)3-s + (0.115 + 0.506i)4-s + (0.593 − 0.404i)5-s + (0.324 − 0.562i)6-s + (0.505 + 0.875i)7-s + (−0.948 − 0.456i)8-s + (−0.117 + 0.0360i)9-s + (−0.0371 + 0.495i)10-s + (0.405 − 1.77i)11-s + (−0.177 − 0.453i)12-s + (−0.100 − 1.34i)13-s + (−0.692 − 0.104i)14-s + (−0.492 + 0.457i)15-s + (0.188 − 0.0909i)16-s + (1.26 + 0.859i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.12591 + 0.422024i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12591 + 0.422024i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.35e7 - 1.78e7i)T \) |
| good | 2 | \( 1 + (9.77 - 12.2i)T + (-113. - 499. i)T^{2} \) |
| 3 | \( 1 + (129. - 19.5i)T + (1.88e4 - 5.80e3i)T^{2} \) |
| 5 | \( 1 + (-828. + 565. i)T + (7.13e5 - 1.81e6i)T^{2} \) |
| 7 | \( 1 + (-3.21e3 - 5.56e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-1.96e4 + 8.62e4i)T + (-2.12e9 - 1.02e9i)T^{2} \) |
| 13 | \( 1 + (1.03e4 + 1.38e5i)T + (-1.04e10 + 1.58e9i)T^{2} \) |
| 17 | \( 1 + (-4.33e5 - 2.95e5i)T + (4.33e10 + 1.10e11i)T^{2} \) |
| 19 | \( 1 + (-2.54e4 - 7.85e3i)T + (2.66e11 + 1.81e11i)T^{2} \) |
| 23 | \( 1 + (4.62e5 + 4.29e5i)T + (1.34e11 + 1.79e12i)T^{2} \) |
| 29 | \( 1 + (-1.71e6 - 2.57e5i)T + (1.38e13 + 4.27e12i)T^{2} \) |
| 31 | \( 1 + (-2.23e6 - 5.69e6i)T + (-1.93e13 + 1.79e13i)T^{2} \) |
| 37 | \( 1 + (2.27e6 - 3.94e6i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (-2.08e7 + 2.61e7i)T + (-7.28e13 - 3.19e14i)T^{2} \) |
| 47 | \( 1 + (1.94e6 + 8.51e6i)T + (-1.00e15 + 4.85e14i)T^{2} \) |
| 53 | \( 1 + (5.89e6 - 7.87e7i)T + (-3.26e15 - 4.91e14i)T^{2} \) |
| 59 | \( 1 + (-5.97e7 + 2.87e7i)T + (5.40e15 - 6.77e15i)T^{2} \) |
| 61 | \( 1 + (-2.34e7 + 5.96e7i)T + (-8.57e15 - 7.95e15i)T^{2} \) |
| 67 | \( 1 + (6.19e7 + 1.91e7i)T + (2.24e16 + 1.53e16i)T^{2} \) |
| 71 | \( 1 + (2.50e8 - 2.32e8i)T + (3.42e15 - 4.57e16i)T^{2} \) |
| 73 | \( 1 + (1.67e7 + 2.23e8i)T + (-5.82e16 + 8.77e15i)T^{2} \) |
| 79 | \( 1 + (-1.20e8 - 2.08e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-6.49e8 + 9.78e7i)T + (1.78e17 - 5.51e16i)T^{2} \) |
| 89 | \( 1 + (5.77e8 - 8.70e7i)T + (3.34e17 - 1.03e17i)T^{2} \) |
| 97 | \( 1 + (-2.59e8 + 1.13e9i)T + (-6.84e17 - 3.29e17i)T^{2} \) |
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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
−14.23384423140329315112304070200, −12.64676872612764470074649265694, −11.79354038320502873017464241911, −10.55392288120939553887137207028, −8.829094131674374703784379606247, −8.079234118274288475494529265374, −5.98870607562439996958263780508, −5.56236421694323506048294908683, −3.11269618676272690065734035208, −0.76983350508033865276136776541,
0.920947097896148655957722634643, 2.13342128949981918308883051519, 4.64293943775337957280312154923, 6.10821728680946196384563782722, 7.24413605892558866084747301897, 9.518225894097677951424378926467, 10.19667930841677889454708305106, 11.44141174821982640535561077721, 12.08913635243122834872168633770, 14.07217092235151590499752161080
