| L(s) = 1 | + (−9.41 − 35.1i)2-s + (55.9 − 32.3i)3-s + (−702. + 405. i)4-s + (−1.55e3 + 415. i)5-s + (−1.66e3 − 1.66e3i)6-s + (−5.03e3 − 3.87e3i)7-s + (7.68e3 + 7.68e3i)8-s + (−7.75e3 + 1.34e4i)9-s + (2.92e4 + 5.06e4i)10-s + (9.76e3 − 3.64e4i)11-s + (−2.62e4 + 4.53e4i)12-s + (1.01e5 + 1.60e4i)13-s + (−8.87e4 + 2.13e5i)14-s + (−7.34e4 + 7.34e4i)15-s + (−9.83e3 + 1.70e4i)16-s + (2.70e5 + 4.67e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.416 − 1.55i)2-s + (0.398 − 0.230i)3-s + (−1.37 + 0.791i)4-s + (−1.11 + 0.297i)5-s + (−0.523 − 0.523i)6-s + (−0.792 − 0.609i)7-s + (0.663 + 0.663i)8-s + (−0.393 + 0.682i)9-s + (0.924 + 1.60i)10-s + (0.201 − 0.750i)11-s + (−0.364 + 0.631i)12-s + (0.987 + 0.155i)13-s + (−0.617 + 1.48i)14-s + (−0.374 + 0.374i)15-s + (−0.0375 + 0.0650i)16-s + (0.784 + 1.35i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.543262 - 0.613618i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.543262 - 0.613618i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (5.03e3 + 3.87e3i)T \) |
| 13 | \( 1 + (-1.01e5 - 1.60e4i)T \) |
| good | 2 | \( 1 + (9.41 + 35.1i)T + (-443. + 256i)T^{2} \) |
| 3 | \( 1 + (-55.9 + 32.3i)T + (9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (1.55e3 - 415. i)T + (1.69e6 - 9.76e5i)T^{2} \) |
| 11 | \( 1 + (-9.76e3 + 3.64e4i)T + (-2.04e9 - 1.17e9i)T^{2} \) |
| 17 | \( 1 + (-2.70e5 - 4.67e5i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (7.77e5 - 2.08e5i)T + (2.79e11 - 1.61e11i)T^{2} \) |
| 23 | \( 1 + (5.28e5 + 3.05e5i)T + (9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 - 4.43e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-1.39e6 + 5.22e6i)T + (-2.28e13 - 1.32e13i)T^{2} \) |
| 37 | \( 1 + (7.75e5 - 2.07e5i)T + (1.12e14 - 6.49e13i)T^{2} \) |
| 41 | \( 1 + (7.66e6 + 7.66e6i)T + 3.27e14iT^{2} \) |
| 43 | \( 1 + 8.75e5iT - 5.02e14T^{2} \) |
| 47 | \( 1 + (1.51e7 + 5.63e7i)T + (-9.69e14 + 5.59e14i)T^{2} \) |
| 53 | \( 1 + (4.25e7 + 7.36e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-1.13e8 - 3.04e7i)T + (7.50e15 + 4.33e15i)T^{2} \) |
| 61 | \( 1 + (-2.90e7 - 1.68e7i)T + (5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (1.06e7 + 2.84e6i)T + (2.35e16 + 1.36e16i)T^{2} \) |
| 71 | \( 1 + (-9.07e7 + 9.07e7i)T - 4.58e16iT^{2} \) |
| 73 | \( 1 + (4.93e7 + 1.32e7i)T + (5.09e16 + 2.94e16i)T^{2} \) |
| 79 | \( 1 + (-9.61e7 + 1.66e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-5.40e7 - 5.40e7i)T + 1.86e17iT^{2} \) |
| 89 | \( 1 + (-2.18e8 - 8.14e8i)T + (-3.03e17 + 1.75e17i)T^{2} \) |
| 97 | \( 1 + (-7.25e8 - 7.25e8i)T + 7.60e17iT^{2} \) |
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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
−11.73535268652641122128936509226, −10.86859523458143910823124293962, −10.21286342265321587448707640855, −8.623324961884518335699116334641, −8.046898745653797225324437047473, −6.33085142708811015732666893534, −3.83862408179968086595480331798, −3.51817235085815711143035522679, −2.01072725793234179690276829085, −0.57954189446966844602603789210,
0.47478672369202701784144229412, 3.13521232350286375850897103576, 4.51237607487509688422468772559, 5.98247471678112162943488141547, 6.94636115825550198592668272082, 8.146071160318401012426489071357, 8.874931620254519182843325655860, 9.736143934792927495120486241229, 11.63449927622340068778958082565, 12.59966784793878330531125538693
