L(s) = 1 | − 3·3-s − 6.58i·5-s + (−15.1 + 10.5i)7-s + 9·9-s − 54.2i·11-s − 40.9i·13-s + 19.7i·15-s − 69.4i·17-s + 160.·19-s + (45.5 − 31.7i)21-s + 87.8i·23-s + 81.6·25-s − 27·27-s − 236.·29-s + 131.·31-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.589i·5-s + (−0.820 + 0.571i)7-s + 0.333·9-s − 1.48i·11-s − 0.874i·13-s + 0.340i·15-s − 0.991i·17-s + 1.93·19-s + (0.473 − 0.330i)21-s + 0.796i·23-s + 0.652·25-s − 0.192·27-s − 1.51·29-s + 0.762·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.060872536\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.060872536\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + (15.1 - 10.5i)T \) |
good | 5 | \( 1 + 6.58iT - 125T^{2} \) |
| 11 | \( 1 + 54.2iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 40.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 69.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 160.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 87.8iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 236.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 131.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 23.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 112. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 194. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 269.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 120.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 338.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 267. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 275. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 270. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 1.23e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 691. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 430.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.22e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 381. iT - 9.12e5T^{2} \) |
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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
−9.214781443874789673625583599816, −8.067080073306673912798268706742, −7.32471079149727200655134995305, −6.18516659644103608793354892447, −5.54794032729458851877388347405, −5.02034437280919498585472583581, −3.47600137110906200605241738647, −2.91039011607997325952039389638, −1.10701856807444778172875214759, −0.32722098804350301668556982437,
1.16928604449123735398583732049, 2.40549267286470241547040756072, 3.64482694198604392030232882238, 4.39684483012836475648994811945, 5.45766891962864349498453727518, 6.44926451345927928052087020016, 7.09652573886791476833897374464, 7.54046553641217675739758774724, 8.984972530753116320035922943762, 9.770167220767540806496658323778