L(s) = 1 | + (−1 − 1.73i)2-s + (2.70 − 4.68i)3-s + (−1.99 + 3.46i)4-s − 10.8·6-s + 7.99·8-s + (−10.1 − 17.5i)9-s + (8.48 − 14.6i)11-s + (10.8 + 18.7i)12-s + (−8 − 13.8i)16-s + (−11.2 + 19.5i)17-s + (−20.3 + 35.1i)18-s + (−18.0 − 31.2i)19-s − 33.9·22-s + (21.6 − 37.5i)24-s + (−12.5 + 21.6i)25-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.902 − 1.56i)3-s + (−0.499 + 0.866i)4-s − 1.80·6-s + 0.999·8-s + (−1.12 − 1.95i)9-s + (0.771 − 1.33i)11-s + (0.902 + 1.56i)12-s + (−0.5 − 0.866i)16-s + (−0.664 + 1.15i)17-s + (−1.12 + 1.95i)18-s + (−0.948 − 1.64i)19-s − 1.54·22-s + (0.902 − 1.56i)24-s + (−0.5 + 0.866i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.314159 + 1.36956i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.314159 + 1.36956i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + 1.73i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-2.70 + 4.68i)T + (-4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-8.48 + 14.6i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 + (11.2 - 19.5i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (18.0 + 31.2i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 15.4T + 1.68e3T^{2} \) |
| 43 | \( 1 - 84.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-1.00 + 1.74i)T + (-1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-31 + 53.6i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + (-38.2 + 66.1i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 147.T + 6.88e3T^{2} \) |
| 89 | \( 1 + (15.6 + 27.0i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 53.5T + 9.40e3T^{2} \) |
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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
−10.89014999124565003430732339288, −9.195152273177202194430942696356, −8.800522074456149057378931291868, −8.029785309274818079696739313484, −7.03410544189207183577445510995, −6.12360196148517919780237212743, −4.01480334682765281455539865725, −2.92166359779082192475793830190, −1.87580264089545757055311430296, −0.65344504883636538613491466518,
2.21199839020322494816763333907, 4.08232030034621089920424513562, 4.52609897844488889967135960509, 5.80956683212554461496867887770, 7.12493401246909389329024337984, 8.150439149771521835785324873662, 8.956582984526743986933064842454, 9.714488324699285592738972910518, 10.14652809478279146893485488629, 11.16405226291701730051911803565