| L(s) = 1 | + (1.86 + 0.724i)2-s + (−2.13 + 2.10i)3-s + (2.95 + 2.69i)4-s + (0.693 + 1.20i)5-s + (−5.50 + 2.38i)6-s + (−0.562 + 0.975i)7-s + (3.54 + 7.17i)8-s + (0.106 − 8.99i)9-s + (0.422 + 2.73i)10-s + (−1.11 + 1.92i)11-s + (−11.9 + 0.463i)12-s + (14.4 − 8.34i)13-s + (−1.75 + 1.41i)14-s + (−4.00 − 1.09i)15-s + (1.42 + 15.9i)16-s − 20.2i·17-s + ⋯ |
| L(s) = 1 | + (0.932 + 0.362i)2-s + (−0.711 + 0.702i)3-s + (0.737 + 0.674i)4-s + (0.138 + 0.240i)5-s + (−0.917 + 0.397i)6-s + (−0.0804 + 0.139i)7-s + (0.443 + 0.896i)8-s + (0.0118 − 0.999i)9-s + (0.0422 + 0.273i)10-s + (−0.101 + 0.175i)11-s + (−0.999 + 0.0386i)12-s + (1.11 − 0.642i)13-s + (−0.125 + 0.100i)14-s + (−0.267 − 0.0733i)15-s + (0.0889 + 0.996i)16-s − 1.19i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.247 - 0.968i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.35167 + 1.04939i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35167 + 1.04939i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.86 - 0.724i)T \) |
| 3 | \( 1 + (2.13 - 2.10i)T \) |
| good | 5 | \( 1 + (-0.693 - 1.20i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (0.562 - 0.975i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (1.11 - 1.92i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-14.4 + 8.34i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 20.2iT - 289T^{2} \) |
| 19 | \( 1 + 21.0iT - 361T^{2} \) |
| 23 | \( 1 + (18.0 - 10.4i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (26.9 - 46.7i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-9.19 - 15.9i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 34.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (15.1 - 8.72i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (44.2 + 25.5i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-32.4 - 18.7i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 52.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + (27.1 + 46.9i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-76.5 - 44.2i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (66.0 - 38.1i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 69.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 82.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-19.8 + 34.3i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-56.9 + 98.6i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 25.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-10.4 + 18.0i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−14.71868213958243511976594828589, −13.60294711498317223357671687367, −12.45605004276606544674411657954, −11.37281044381411192279620749251, −10.52001410486758793497407075633, −8.916109856827939551174455847242, −7.12731988868501677005878921391, −5.92479473805803941266840580620, −4.82702959575641360025595694015, −3.26793398941810253841955953628,
1.66992411347533449852005750213, 4.02260627027423274481176974270, 5.70305557939496805514415493007, 6.49429156195486773711580958935, 8.101345882647543410379049131881, 10.11041007508615402214154562798, 11.17687594942770349461877429607, 12.06010441116027381208168695372, 13.12532262148105419087561371154, 13.72497350632842756937807023593
