| L(s) = 1 | + (7.87 + 13.6i)2-s + (40.5 + 23.3i)3-s + (4.09 − 7.09i)4-s + (−872. + 503. i)5-s + 736. i·6-s + (−1.32e3 + 2.00e3i)7-s + 4.15e3·8-s + (1.09e3 + 1.89e3i)9-s + (−1.37e4 − 7.93e3i)10-s + (−9.28e3 + 1.60e4i)11-s + (331. − 191. i)12-s − 1.74e4i·13-s + (−3.77e4 − 2.31e3i)14-s − 4.71e4·15-s + (3.16e4 + 5.48e4i)16-s + (−2.57e4 − 1.48e4i)17-s + ⋯ |
| L(s) = 1 | + (0.491 + 0.852i)2-s + (0.5 + 0.288i)3-s + (0.0159 − 0.0276i)4-s + (−1.39 + 0.806i)5-s + 0.568i·6-s + (−0.552 + 0.833i)7-s + 1.01·8-s + (0.166 + 0.288i)9-s + (−1.37 − 0.793i)10-s + (−0.633 + 1.09i)11-s + (0.0159 − 0.00923i)12-s − 0.611i·13-s + (−0.982 − 0.0601i)14-s − 0.930·15-s + (0.483 + 0.837i)16-s + (−0.308 − 0.178i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.442i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.412393 + 1.76908i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.412393 + 1.76908i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-40.5 - 23.3i)T \) |
| 7 | \( 1 + (1.32e3 - 2.00e3i)T \) |
| good | 2 | \( 1 + (-7.87 - 13.6i)T + (-128 + 221. i)T^{2} \) |
| 5 | \( 1 + (872. - 503. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (9.28e3 - 1.60e4i)T + (-1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + 1.74e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (2.57e4 + 1.48e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-1.19e5 + 6.87e4i)T + (8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-1.86e5 - 3.22e5i)T + (-3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 - 1.12e6T + 5.00e11T^{2} \) |
| 31 | \( 1 + (4.08e5 + 2.36e5i)T + (4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-1.30e6 - 2.26e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 - 3.18e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 4.06e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (3.90e5 - 2.25e5i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-5.50e6 + 9.53e6i)T + (-3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (5.81e6 + 3.35e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-4.12e6 + 2.38e6i)T + (9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (4.32e6 - 7.48e6i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 4.14e6T + 6.45e14T^{2} \) |
| 73 | \( 1 + (2.08e7 + 1.20e7i)T + (4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (1.15e7 + 1.99e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 - 6.29e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-2.43e7 + 1.40e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 - 7.60e7iT - 7.83e15T^{2} \) |
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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
−16.02417200326938527077424290234, −15.36215596506257192656185772495, −14.91243758029716925700056557665, −13.22466004358226747524637219321, −11.56934886331693545354568815016, −10.00920043544525861256365602939, −7.950256625353605030939767681582, −6.89078528365057217475621237439, −4.90451984577133264199041394537, −3.03059504234991800259725148308,
0.76206338658525279302640389267, 3.19218146574058159928426590787, 4.33366838292423950662706734366, 7.33599113554345516095122349206, 8.531871838257564190751211183059, 10.69292333820489008577214562454, 11.95274114452855233420807454582, 12.92159205590963806836545218330, 13.97339083046658961129538789579, 15.98816135714785746687653212593
