| L(s) = 1 | + (3.91 + 2.25i)2-s + (21.9 − 37.9i)3-s + (−53.7 − 93.1i)4-s − 49.0i·5-s + (171. − 99.0i)6-s + (736. − 425. i)7-s − 1.06e3i·8-s + (132. + 229. i)9-s + (110. − 191. i)10-s + (−108. − 62.4i)11-s − 4.71e3·12-s + (−784. + 7.88e3i)13-s + 3.84e3·14-s + (−1.86e3 − 1.07e3i)15-s + (−4.48e3 + 7.76e3i)16-s + (5.66e3 + 9.80e3i)17-s + ⋯ |
| L(s) = 1 | + (0.345 + 0.199i)2-s + (0.468 − 0.811i)3-s + (−0.420 − 0.727i)4-s − 0.175i·5-s + (0.324 − 0.187i)6-s + (0.811 − 0.468i)7-s − 0.734i·8-s + (0.0606 + 0.104i)9-s + (0.0350 − 0.0606i)10-s + (−0.0244 − 0.0141i)11-s − 0.787·12-s + (−0.0990 + 0.995i)13-s + 0.374·14-s + (−0.142 − 0.0822i)15-s + (−0.273 + 0.473i)16-s + (0.279 + 0.484i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.423 + 0.905i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.423 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.59782 - 1.01696i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59782 - 1.01696i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (784. - 7.88e3i)T \) |
| good | 2 | \( 1 + (-3.91 - 2.25i)T + (64 + 110. i)T^{2} \) |
| 3 | \( 1 + (-21.9 + 37.9i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + 49.0iT - 7.81e4T^{2} \) |
| 7 | \( 1 + (-736. + 425. i)T + (4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (108. + 62.4i)T + (9.74e6 + 1.68e7i)T^{2} \) |
| 17 | \( 1 + (-5.66e3 - 9.80e3i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (6.81e3 - 3.93e3i)T + (4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-4.31e4 + 7.46e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (6.28e4 - 1.08e5i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 - 7.88e4iT - 2.75e10T^{2} \) |
| 37 | \( 1 + (-1.45e5 - 8.40e4i)T + (4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (3.00e5 + 1.73e5i)T + (9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (2.43e5 + 4.21e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + 4.43e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 1.44e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (2.30e6 - 1.33e6i)T + (1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-6.48e5 - 1.12e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-4.03e6 - 2.32e6i)T + (3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (3.54e6 - 2.04e6i)T + (4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + 6.09e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 1.13e3T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.26e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + (1.63e6 + 9.46e5i)T + (2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (1.45e7 - 8.41e6i)T + (4.03e13 - 6.99e13i)T^{2} \) |
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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
−18.37514440342157597222315697499, −16.68414198419128951646669308289, −14.74341642421490598820437954228, −13.94348863539193007340125510039, −12.71068092171795610416255171420, −10.63716743357654771960570093380, −8.658887053981933642540751284941, −6.89712862671303288638216573306, −4.71835262754101437769631399076, −1.44004529894499807317708428913,
3.16638629415429897230058103301, 4.90331677949212270497649057693, 7.989319770967926995916016762226, 9.427913938638006346747911328312, 11.33306673332854744434030046616, 12.87842962375971588920510294928, 14.45464154731911769526855464661, 15.48311956970013184666117243085, 17.21567871068271622398571143806, 18.37178223155313249242805627701
