| L(s) = 1 | + (−2.74 + 4.75i)2-s + (−1.70 − 2.94i)3-s + (240. + 417. i)4-s + (−828. + 1.43e3i)5-s + 18.6·6-s + (2.82e3 + 5.69e3i)7-s − 5.45e3·8-s + (9.83e3 − 1.70e4i)9-s + (−4.54e3 − 7.87e3i)10-s + (−1.68e4 − 2.91e4i)11-s + (820. − 1.42e3i)12-s + 4.79e4·13-s + (−3.47e4 − 2.20e3i)14-s + 5.64e3·15-s + (−1.08e5 + 1.87e5i)16-s + (1.72e5 + 2.99e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.121 + 0.210i)2-s + (−0.0121 − 0.0210i)3-s + (0.470 + 0.815i)4-s + (−0.593 + 1.02i)5-s + 0.00588·6-s + (0.444 + 0.895i)7-s − 0.470·8-s + (0.499 − 0.865i)9-s + (−0.143 − 0.249i)10-s + (−0.346 − 0.600i)11-s + (0.0114 − 0.0197i)12-s + 0.465·13-s + (−0.242 − 0.0153i)14-s + 0.0288·15-s + (−0.413 + 0.716i)16-s + (0.502 + 0.869i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.000219 - 0.999i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.000219 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.962071 + 0.962282i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.962071 + 0.962282i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.82e3 - 5.69e3i)T \) |
| good | 2 | \( 1 + (2.74 - 4.75i)T + (-256 - 443. i)T^{2} \) |
| 3 | \( 1 + (1.70 + 2.94i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (828. - 1.43e3i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 11 | \( 1 + (1.68e4 + 2.91e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 - 4.79e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + (-1.72e5 - 2.99e5i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-2.02e5 + 3.50e5i)T + (-1.61e11 - 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-9.26e5 + 1.60e6i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 - 6.82e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-4.54e6 - 7.87e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-7.77e6 + 1.34e7i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + 2.98e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 6.28e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + (5.16e6 - 8.93e6i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (3.32e7 + 5.75e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (3.52e7 + 6.11e7i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-4.21e7 + 7.30e7i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.05e8 - 1.83e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 - 2.31e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + (1.24e8 + 2.15e8i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (1.33e8 - 2.30e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 - 6.33e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (3.11e8 - 5.38e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + 9.94e8T + 7.60e17T^{2} \) |
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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
−20.93136809069602134418584109765, −18.90100800902004101758284723359, −17.83018431176297885882959290598, −15.89913772913792454212114134572, −14.87531363141998757822036095919, −12.42607014030712190759746878495, −11.08396849916735373559901030186, −8.410426124784862540419246889968, −6.66734352418330640950634354107, −3.17932847120696140692856041184,
1.18089554118922301207582774195, 4.86994364224670782644975698177, 7.66272054885647040106170760717, 10.00950153722469222705011779844, 11.59799708554619691838692389789, 13.55296672243535620254730089741, 15.48662234571430652597936980886, 16.70275458677987113424494062811, 18.72995575333795588436883095465, 20.11453679213555019878465768203
