L(s) = 1 | + (10.0 + 3.64i)5-s + (−2.90 + 16.4i)7-s + (−3.53 + 1.28i)11-s + (−55.0 + 46.1i)13-s + (14.0 + 24.3i)17-s + (4.34 − 7.52i)19-s + (−4.21 − 23.9i)23-s + (−8.84 − 7.42i)25-s + (−183. − 153. i)29-s + (45.3 + 257. i)31-s + (−89.2 + 154. i)35-s + (50.0 + 86.6i)37-s + (−177. + 149. i)41-s + (220. − 80.1i)43-s + (−98.9 + 561. i)47-s + ⋯ |
L(s) = 1 | + (0.895 + 0.325i)5-s + (−0.157 + 0.890i)7-s + (−0.0969 + 0.0352i)11-s + (−1.17 + 0.985i)13-s + (0.200 + 0.347i)17-s + (0.0524 − 0.0908i)19-s + (−0.0382 − 0.216i)23-s + (−0.0707 − 0.0593i)25-s + (−1.17 − 0.983i)29-s + (0.262 + 1.49i)31-s + (−0.430 + 0.746i)35-s + (0.222 + 0.385i)37-s + (−0.676 + 0.567i)41-s + (0.781 − 0.284i)43-s + (−0.307 + 1.74i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.454125968\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.454125968\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-10.0 - 3.64i)T + (95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (2.90 - 16.4i)T + (-322. - 117. i)T^{2} \) |
| 11 | \( 1 + (3.53 - 1.28i)T + (1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (55.0 - 46.1i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-14.0 - 24.3i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-4.34 + 7.52i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (4.21 + 23.9i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (183. + 153. i)T + (4.23e3 + 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-45.3 - 257. i)T + (-2.79e4 + 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-50.0 - 86.6i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (177. - 149. i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-220. + 80.1i)T + (6.09e4 - 5.11e4i)T^{2} \) |
| 47 | \( 1 + (98.9 - 561. i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 + 368.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-485. - 176. i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (106. - 605. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (264. - 221. i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (496. + 860. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (180. - 312. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-127. - 107. i)T + (8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-718. - 602. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-404. + 700. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-1.72e3 + 627. i)T + (6.99e5 - 5.86e5i)T^{2} \) |
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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
−11.61106397356347853727730882515, −10.38826129545416990642862512347, −9.607302678348329363882825827331, −8.913301435057783794362221979400, −7.61536631046582023876556591456, −6.49628758824857252502189977270, −5.69029106756110078722374927173, −4.54000011581618714968947979250, −2.82703265220755932420869619049, −1.86810995677106013727108437565,
0.49209724978456709173620084685, 2.09068306885166226243582195871, 3.54907773376351771785457050869, 4.98494482236257972214459664097, 5.78854637840905455600992560400, 7.12527654274071976755195636479, 7.86193974002579877113312802747, 9.263327800049530553367004896054, 9.923685253935838891863921061100, 10.66590200573752564788938680405