L(s) = 1 | − 9.36i·2-s + (26.7 + 3.90i)3-s − 23.6·4-s − 36.7i·5-s + (36.5 − 250. i)6-s + 305.·7-s − 377. i·8-s + (698. + 208. i)9-s − 344.·10-s − 401. i·11-s + (−631. − 92.1i)12-s − 1.88e3·13-s − 2.85e3i·14-s + (143. − 982. i)15-s − 5.04e3·16-s − 2.57e3i·17-s + ⋯ |
L(s) = 1 | − 1.17i·2-s + (0.989 + 0.144i)3-s − 0.369·4-s − 0.294i·5-s + (0.169 − 1.15i)6-s + 0.889·7-s − 0.738i·8-s + (0.958 + 0.285i)9-s − 0.344·10-s − 0.301i·11-s + (−0.365 − 0.0533i)12-s − 0.858·13-s − 1.04i·14-s + (0.0424 − 0.291i)15-s − 1.23·16-s − 0.524i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.144 + 0.989i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.144 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.63321 - 1.88895i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.63321 - 1.88895i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-26.7 - 3.90i)T \) |
| 11 | \( 1 + 401. iT \) |
good | 2 | \( 1 + 9.36iT - 64T^{2} \) |
| 5 | \( 1 + 36.7iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 305.T + 1.17e5T^{2} \) |
| 13 | \( 1 + 1.88e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 2.57e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 2.65e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 1.77e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 1.29e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 2.43e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 4.81e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 4.61e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 6.39e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.52e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 2.06e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 3.31e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 6.98e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 4.88e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 3.38e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 5.06e4T + 1.51e11T^{2} \) |
| 79 | \( 1 - 1.21e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 1.02e6iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 7.19e4iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 6.93e5T + 8.32e11T^{2} \) |
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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
−14.88867264858148615401376629534, −13.72442667507146111702950945594, −12.54506893771940804903193917249, −11.33931292523265115898315152160, −10.07008401109606485748236549885, −8.892240346729432429282810837603, −7.40096048845666600616198452377, −4.61922054013158638434995547727, −2.95235980899489314431909913171, −1.47368440862309730146660697527,
2.28408310796714170436251248548, 4.69313978949970362035466829601, 6.67200088704738303237355328489, 7.78259682785751424904445759431, 8.748460222484068345982674213013, 10.53494884710556019237868244971, 12.31951825120971973076678129195, 13.97470725990932205823221079126, 14.73310325279644800015322979247, 15.37095285658642287247612516806