| L(s) = 1 | + (−7.27 − 27.1i)2-s + (−177. + 102. i)3-s + (−241. + 139. i)4-s + (171. − 45.8i)5-s + (4.06e3 + 4.06e3i)6-s + (−5.80e3 − 2.56e3i)7-s + (−4.63e3 − 4.63e3i)8-s + (1.10e4 − 1.91e4i)9-s + (−2.49e3 − 4.31e3i)10-s + (−1.03e4 + 3.86e4i)11-s + (2.84e4 − 4.93e4i)12-s + (3.61e4 + 9.64e4i)13-s + (−2.74e4 + 1.76e5i)14-s + (−2.56e4 + 2.56e4i)15-s + (−1.63e5 + 2.83e5i)16-s + (2.78e4 + 4.82e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.321 − 1.20i)2-s + (−1.26 + 0.728i)3-s + (−0.471 + 0.272i)4-s + (0.122 − 0.0328i)5-s + (1.28 + 1.28i)6-s + (−0.914 − 0.404i)7-s + (−0.400 − 0.400i)8-s + (0.561 − 0.972i)9-s + (−0.0787 − 0.136i)10-s + (−0.213 + 0.795i)11-s + (0.396 − 0.687i)12-s + (0.351 + 0.936i)13-s + (−0.191 + 1.22i)14-s + (−0.130 + 0.130i)15-s + (−0.624 + 1.08i)16-s + (0.0809 + 0.140i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.806 + 0.591i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.806 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.121229 - 0.369936i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.121229 - 0.369936i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (5.80e3 + 2.56e3i)T \) |
| 13 | \( 1 + (-3.61e4 - 9.64e4i)T \) |
| good | 2 | \( 1 + (7.27 + 27.1i)T + (-443. + 256i)T^{2} \) |
| 3 | \( 1 + (177. - 102. i)T + (9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-171. + 45.8i)T + (1.69e6 - 9.76e5i)T^{2} \) |
| 11 | \( 1 + (1.03e4 - 3.86e4i)T + (-2.04e9 - 1.17e9i)T^{2} \) |
| 17 | \( 1 + (-2.78e4 - 4.82e4i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-2.66e5 + 7.14e4i)T + (2.79e11 - 1.61e11i)T^{2} \) |
| 23 | \( 1 + (-1.78e6 - 1.02e6i)T + (9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + 1.72e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-3.43e4 + 1.28e5i)T + (-2.28e13 - 1.32e13i)T^{2} \) |
| 37 | \( 1 + (-4.61e6 + 1.23e6i)T + (1.12e14 - 6.49e13i)T^{2} \) |
| 41 | \( 1 + (4.86e6 + 4.86e6i)T + 3.27e14iT^{2} \) |
| 43 | \( 1 - 1.69e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 + (-1.53e6 - 5.74e6i)T + (-9.69e14 + 5.59e14i)T^{2} \) |
| 53 | \( 1 + (3.49e7 + 6.05e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (9.94e7 + 2.66e7i)T + (7.50e15 + 4.33e15i)T^{2} \) |
| 61 | \( 1 + (2.58e7 + 1.49e7i)T + (5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (5.41e7 + 1.44e7i)T + (2.35e16 + 1.36e16i)T^{2} \) |
| 71 | \( 1 + (-7.68e7 + 7.68e7i)T - 4.58e16iT^{2} \) |
| 73 | \( 1 + (2.32e8 + 6.23e7i)T + (5.09e16 + 2.94e16i)T^{2} \) |
| 79 | \( 1 + (-4.93e7 + 8.53e7i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + (4.66e7 + 4.66e7i)T + 1.86e17iT^{2} \) |
| 89 | \( 1 + (7.58e7 + 2.83e8i)T + (-3.03e17 + 1.75e17i)T^{2} \) |
| 97 | \( 1 + (8.60e8 + 8.60e8i)T + 7.60e17iT^{2} \) |
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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
−11.49070618665712729161425546855, −10.88353825893339896188946959222, −9.806186170744689097748992114571, −9.387576976545510985824653021156, −7.03212231599294748749476217482, −5.90852612579489958477686628624, −4.47419846399089827926954503832, −3.29910076343761371118006075198, −1.56572430902694023174346732741, −0.21204632044442987361246117606,
0.75021248475946696179939576614, 2.92055645991935925446681842349, 5.38210168609561828802535255891, 5.99234639197159617522224509444, 6.78747881709745800760779296232, 7.88328354083251451105977557583, 9.109528394970948201730050350804, 10.62055185694345982223575542742, 11.71543049981717347212886376025, 12.66903981615064335187091799560
