L(s) = 1 | + (−5.5 + 9.52i)7-s + (−11.5 − 19.9i)13-s − 37·19-s + (−12.5 + 21.6i)25-s + (23 + 39.8i)31-s − 73·37-s + (11 − 19.0i)43-s + (−36 − 62.3i)49-s + (−23.5 + 40.7i)61-s + (6.5 + 11.2i)67-s + 143·73-s + (−5.5 + 9.52i)79-s + 253·91-s + (84.5 − 146. i)97-s + (78.5 + 135. i)103-s + ⋯ |
L(s) = 1 | + (−0.785 + 1.36i)7-s + (−0.884 − 1.53i)13-s − 1.94·19-s + (−0.5 + 0.866i)25-s + (0.741 + 1.28i)31-s − 1.97·37-s + (0.255 − 0.443i)43-s + (−0.734 − 1.27i)49-s + (−0.385 + 0.667i)61-s + (0.0970 + 0.168i)67-s + 1.95·73-s + (−0.0696 + 0.120i)79-s + 2.78·91-s + (0.871 − 1.50i)97-s + (0.762 + 1.32i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0263584 + 0.301278i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0263584 + 0.301278i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (5.5 - 9.52i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (11.5 + 19.9i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 + 37T + 361T^{2} \) |
| 23 | \( 1 + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-23 - 39.8i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 73T + 1.36e3T^{2} \) |
| 41 | \( 1 + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-11 + 19.0i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (23.5 - 40.7i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-6.5 - 11.2i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 143T + 5.32e3T^{2} \) |
| 79 | \( 1 + (5.5 - 9.52i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + (-84.5 + 146. i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−12.19787016526518731289295178227, −10.75836997100981482954025134063, −10.01742406937010054516506518100, −8.956745681547651525506563841765, −8.242177038758144632334732255818, −6.92837239049983512834012919640, −5.89166597756973500113469119646, −5.04188097132166076130779560552, −3.35090367058798269526636706664, −2.28677759127080326458757997554,
0.13090030908602776820523889904, 2.15353844291880143779285500069, 3.87126525386735011037495887685, 4.57565952732927872104641596823, 6.38421989711552048256758718644, 6.89405383267080586217379509987, 8.029560461999959198162040234432, 9.240742061622227408066858555585, 10.07161338009180343209276048616, 10.80692979765177978195100061491