L(s) = 1 | + (13.5 + 23.3i)3-s + (−50 + 86.6i)5-s + (−364.5 + 631. i)9-s + (−1.38e3 − 2.40e3i)11-s − 3.29e3·13-s − 2.70e3·15-s + (−2.95e3 − 5.10e3i)17-s + (−3.32e3 + 5.75e3i)19-s + (−991 + 1.71e3i)23-s + (3.40e4 + 5.89e4i)25-s − 1.96e4·27-s − 2.08e5·29-s + (5.88e4 + 1.02e5i)31-s + (3.74e4 − 6.48e4i)33-s + (1.67e5 − 2.90e5i)37-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (−0.178 + 0.309i)5-s + (−0.166 + 0.288i)9-s + (−0.314 − 0.544i)11-s − 0.415·13-s − 0.206·15-s + (−0.145 − 0.252i)17-s + (−0.111 + 0.192i)19-s + (−0.0169 + 0.0294i)23-s + (0.435 + 0.755i)25-s − 0.192·27-s − 1.58·29-s + (0.355 + 0.615i)31-s + (0.181 − 0.314i)33-s + (0.544 − 0.943i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(1.674718660\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.674718660\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-13.5 - 23.3i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (50 - 86.6i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (1.38e3 + 2.40e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + 3.29e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + (2.95e3 + 5.10e3i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (3.32e3 - 5.75e3i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (991 - 1.71e3i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + 2.08e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + (-5.88e4 - 1.02e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-1.67e5 + 2.90e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + 2.65e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.32e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-3.28e5 + 5.69e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-3.04e5 - 5.27e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-2.68e5 - 4.64e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-8.98e5 + 1.55e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.06e6 + 1.83e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + 1.19e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (5.28e5 + 9.14e5i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (4.99e5 - 8.64e5i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 - 3.89e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-2.31e6 + 4.00e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 - 1.52e7T + 8.07e13T^{2} \) |
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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
−9.495818987264873884522411299940, −8.785263276893203329811772806620, −7.79158005181949934800826407394, −7.02072238307116102495631711165, −5.80497468872726357241131205572, −4.96147929170139513697898724779, −3.82208079131516615931526356892, −3.02111947435136813503268888911, −1.92356497833029215926701013168, −0.41759672478793209281249335355,
0.67074930057982823709831660355, 1.85958056382846429834314100631, 2.75950054500881910774087486123, 4.02602473199963266099005443967, 4.95749975630065742813192181952, 6.06285218555691671597483061057, 7.05102207197775877800884172710, 7.81551376058665930169679895286, 8.616739772922722157718291234629, 9.526310789008828306810841051286