| L(s) = 1 | + (−0.707 + 1.22i)2-s + (−0.999 − 1.73i)4-s + (1.29 − 4.82i)5-s + (3.46 + 2i)7-s + 2.82·8-s + (5 + 5i)10-s + (−9.79 − 5.65i)11-s + (−15.5 + 9i)13-s + (−4.89 + 2.82i)14-s + (−2.00 + 3.46i)16-s + 1.41·17-s − 24·19-s + (−9.65 + 2.58i)20-s + (13.8 − 7.99i)22-s + (19.7 + 34.2i)23-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.258 − 0.965i)5-s + (0.494 + 0.285i)7-s + 0.353·8-s + (0.5 + 0.5i)10-s + (−0.890 − 0.514i)11-s + (−1.19 + 0.692i)13-s + (−0.349 + 0.202i)14-s + (−0.125 + 0.216i)16-s + 0.0831·17-s − 1.26·19-s + (−0.482 + 0.129i)20-s + (0.629 − 0.363i)22-s + (0.860 + 1.49i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 - 0.819i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8234117973\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8234117973\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.707 - 1.22i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.29 + 4.82i)T \) |
| good | 7 | \( 1 + (-3.46 - 2i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (9.79 + 5.65i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (15.5 - 9i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 1.41T + 289T^{2} \) |
| 19 | \( 1 + 24T + 361T^{2} \) |
| 23 | \( 1 + (-19.7 - 34.2i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-33.0 - 19.0i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 56iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-20.8 + 12.0i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-69.2 - 40i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (14.1 - 24.4i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 4.24T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-53.8 + 31.1i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (55 - 95.2i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (27.7 - 16i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 50.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 46iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (18 - 31.1i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-2.82 + 4.89i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 57.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-12.1 - 7i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.10506717922859952680537360266, −9.331029160399032328981639908161, −8.586673974509257042530082817108, −7.937224629194500844971434241662, −7.00044507757271926818242359787, −5.86309081404528642139169537361, −5.08180354087901850360874347487, −4.42379481772891468556294884343, −2.59210392717624246448333189418, −1.27775750217808137571923147199,
0.32117540850158209047331599691, 2.24825215764796783659711798668, 2.73595904892477894041227557339, 4.22037490337718639518186998988, 5.11139957659006667000616389694, 6.40172278332897614706692011070, 7.39837191163518153745635007437, 7.951106992198720486634472527371, 9.049618723189248973318621354265, 10.08406375837613175590416724225
