L(s) = 1 | + (1.18 − 2.75i)3-s + (0.00985 + 0.00985i)5-s + 6.42i·7-s + (−6.19 − 6.53i)9-s + (−9.07 − 9.07i)11-s + (−12.6 − 12.6i)13-s + (0.0388 − 0.0154i)15-s − 19.0i·17-s + (2.07 + 2.07i)19-s + (17.7 + 7.61i)21-s + 19.5·23-s − 24.9i·25-s + (−25.3 + 9.32i)27-s + (11.1 − 11.1i)29-s − 59.9·31-s + ⋯ |
L(s) = 1 | + (0.395 − 0.918i)3-s + (0.00197 + 0.00197i)5-s + 0.917i·7-s + (−0.687 − 0.725i)9-s + (−0.824 − 0.824i)11-s + (−0.969 − 0.969i)13-s + (0.00259 − 0.00103i)15-s − 1.11i·17-s + (0.109 + 0.109i)19-s + (0.842 + 0.362i)21-s + 0.850·23-s − 0.999i·25-s + (−0.938 + 0.345i)27-s + (0.385 − 0.385i)29-s − 1.93·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.349567 - 1.10033i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.349567 - 1.10033i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.18 + 2.75i)T \) |
good | 5 | \( 1 + (-0.00985 - 0.00985i)T + 25iT^{2} \) |
| 7 | \( 1 - 6.42iT - 49T^{2} \) |
| 11 | \( 1 + (9.07 + 9.07i)T + 121iT^{2} \) |
| 13 | \( 1 + (12.6 + 12.6i)T + 169iT^{2} \) |
| 17 | \( 1 + 19.0iT - 289T^{2} \) |
| 19 | \( 1 + (-2.07 - 2.07i)T + 361iT^{2} \) |
| 23 | \( 1 - 19.5T + 529T^{2} \) |
| 29 | \( 1 + (-11.1 + 11.1i)T - 841iT^{2} \) |
| 31 | \( 1 + 59.9T + 961T^{2} \) |
| 37 | \( 1 + (9.32 - 9.32i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 47.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + (24.1 - 24.1i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + 6.29iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (20.6 + 20.6i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-60.3 - 60.3i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (48.0 + 48.0i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (-23.7 - 23.7i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 13.5T + 5.04e3T^{2} \) |
| 73 | \( 1 - 31.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 47.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-70.3 + 70.3i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 95.1T + 7.92e3T^{2} \) |
| 97 | \( 1 - 61.6T + 9.40e3T^{2} \) |
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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
−10.90066525509157575159913511877, −9.667171425409265426950163824222, −8.764015200422532230393322402413, −7.940142398170613286626574435839, −7.14333198689196808778634817473, −5.87889828588839227935042213374, −5.13606252745932622255223443921, −3.09828348397767788337285171646, −2.40466137976991044871459715391, −0.45618161345273212472290891794,
2.06476238451880831744310427836, 3.54365909742241215897802582778, 4.51603158673262687665413061236, 5.37346715829657319772391259493, 7.03377800769259692897988465049, 7.69212639411816978145155147904, 8.965155014584971776554962785446, 9.680376208445265750740969154595, 10.57068287398547221315107392588, 11.10598120334341370766343181884