L(s) = 1 | + (0.532 − 1.98i)2-s + (−0.220 + 5.19i)3-s + (3.26 + 1.88i)4-s + (4.41 + 4.41i)5-s + (10.1 + 3.20i)6-s + (3.85 − 1.03i)7-s + (17.1 − 17.1i)8-s + (−26.9 − 2.29i)9-s + (11.1 − 6.41i)10-s + (5.64 + 1.51i)11-s + (−10.5 + 16.5i)12-s + (−11.9 − 45.3i)13-s − 8.21i·14-s + (−23.8 + 21.9i)15-s + (−9.78 − 16.9i)16-s + (−53.6 + 92.8i)17-s + ⋯ |
L(s) = 1 | + (0.188 − 0.702i)2-s + (−0.0424 + 0.999i)3-s + (0.408 + 0.235i)4-s + (0.394 + 0.394i)5-s + (0.693 + 0.217i)6-s + (0.208 − 0.0558i)7-s + (0.756 − 0.756i)8-s + (−0.996 − 0.0848i)9-s + (0.351 − 0.203i)10-s + (0.154 + 0.0414i)11-s + (−0.252 + 0.398i)12-s + (−0.254 − 0.967i)13-s − 0.156i·14-s + (−0.411 + 0.377i)15-s + (−0.152 − 0.264i)16-s + (−0.764 + 1.32i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.59920 + 0.151816i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.59920 + 0.151816i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.220 - 5.19i)T \) |
| 13 | \( 1 + (11.9 + 45.3i)T \) |
good | 2 | \( 1 + (-0.532 + 1.98i)T + (-6.92 - 4i)T^{2} \) |
| 5 | \( 1 + (-4.41 - 4.41i)T + 125iT^{2} \) |
| 7 | \( 1 + (-3.85 + 1.03i)T + (297. - 171.5i)T^{2} \) |
| 11 | \( 1 + (-5.64 - 1.51i)T + (1.15e3 + 665.5i)T^{2} \) |
| 17 | \( 1 + (53.6 - 92.8i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (15.7 + 58.6i)T + (-5.94e3 + 3.42e3i)T^{2} \) |
| 23 | \( 1 + (56.5 + 97.9i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-118. + 68.6i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (10.7 - 10.7i)T - 2.97e4iT^{2} \) |
| 37 | \( 1 + (55.7 - 208. i)T + (-4.38e4 - 2.53e4i)T^{2} \) |
| 41 | \( 1 + (83.3 - 311. i)T + (-5.96e4 - 3.44e4i)T^{2} \) |
| 43 | \( 1 + (-62.0 - 35.7i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-258. + 258. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 + 216. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-166. - 621. i)T + (-1.77e5 + 1.02e5i)T^{2} \) |
| 61 | \( 1 + (371. - 644. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (335. + 90.0i)T + (2.60e5 + 1.50e5i)T^{2} \) |
| 71 | \( 1 + (347. - 93.1i)T + (3.09e5 - 1.78e5i)T^{2} \) |
| 73 | \( 1 + (-761. - 761. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 - 670.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (1.03e3 + 1.03e3i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + (-841. - 225. i)T + (6.10e5 + 3.52e5i)T^{2} \) |
| 97 | \( 1 + (-9.67 - 36.1i)T + (-7.90e5 + 4.56e5i)T^{2} \) |
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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
−15.66655112299543981858602555437, −14.75502444618454132599488701027, −13.29631398694305243340997127490, −11.95574648386834281416206005300, −10.70660770169757169854301893890, −10.15669774358302499471312385255, −8.346364341307225988097854445849, −6.35055688945708062474213907783, −4.35276112940722624457406592367, −2.69869947798531341196252478966,
1.89449762321366512715333234970, 5.24267197064491767173598717110, 6.55207455131161802699024625140, 7.60786528663877428787944465303, 9.153228513307208523153129960695, 11.12750315344822128077514856371, 12.17617331468862725262073578195, 13.71224039184808402348420684646, 14.28324430766890462118780718054, 15.78706061495352454723535220487