L(s) = 1 | + (1 − 1.73i)2-s + (4.5 + 7.79i)3-s + (14 + 24.2i)4-s + (−5.5 + 9.52i)5-s + 18·6-s + (129.5 + 6.06i)7-s + 120·8-s + (−40.5 + 70.1i)9-s + (11 + 19.0i)10-s + (−134.5 − 232. i)11-s + (−126 + 218. i)12-s − 308·13-s + (140 − 218. i)14-s − 99·15-s + (−328 + 568. i)16-s + (−948 − 1.64e3i)17-s + ⋯ |
L(s) = 1 | + (0.176 − 0.306i)2-s + (0.288 + 0.499i)3-s + (0.437 + 0.757i)4-s + (−0.0983 + 0.170i)5-s + 0.204·6-s + (0.998 + 0.0467i)7-s + 0.662·8-s + (−0.166 + 0.288i)9-s + (0.0347 + 0.0602i)10-s + (−0.335 − 0.580i)11-s + (−0.252 + 0.437i)12-s − 0.505·13-s + (0.190 − 0.297i)14-s − 0.113·15-s + (−0.320 + 0.554i)16-s + (−0.795 − 1.37i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.77862 + 0.492437i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.77862 + 0.492437i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-4.5 - 7.79i)T \) |
| 7 | \( 1 + (-129.5 - 6.06i)T \) |
good | 2 | \( 1 + (-1 + 1.73i)T + (-16 - 27.7i)T^{2} \) |
| 5 | \( 1 + (5.5 - 9.52i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (134.5 + 232. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + 308T + 3.71e5T^{2} \) |
| 17 | \( 1 + (948 + 1.64e3i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-82 + 142. i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-1.63e3 + 2.82e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 - 2.41e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (1.42e3 + 2.46e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-5.66e3 + 9.81e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + 1.68e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.89e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.05e4 - 1.82e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-1.48e4 - 2.57e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-4.08e3 - 7.06e3i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (7.58e3 - 1.31e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-1.60e4 - 2.77e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 3.82e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (1.74e4 + 3.01e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (6.76e3 - 1.17e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 6.81e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-5.74e4 + 9.95e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 1.54e5T + 8.58e9T^{2} \) |
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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
−17.07992254346787361207996444366, −15.96637152045561049916619260318, −14.67068236937261504725333614358, −13.32028728743162366899138785025, −11.69275279578636196646526949339, −10.78221884022388611520688050872, −8.754702276809893504435825252344, −7.35144283289989612574614320361, −4.69903887003035454673668142909, −2.72825060340540242553584079017,
1.75412286179518563771849019185, 4.97554966483568235792382431637, 6.81750658270653134090017068062, 8.270496920005879449165935584238, 10.26754123727480900395552315345, 11.68393884921847301776416933191, 13.28487970260665828998566639764, 14.64668832105816462542178637351, 15.34398538378447585864942778982, 17.05913613541998908204235982049