L(s) = 1 | + (2 − 3.46i)2-s + (−3.99 − 6.92i)4-s + (2 − 3.46i)5-s + (−7.99 − 13.8i)10-s + (31 + 53.6i)11-s + 62·13-s + (31.9 − 55.4i)16-s + (−42 − 72.7i)17-s + (50 − 86.6i)19-s − 31.9·20-s + 248·22-s + (−21 + 36.3i)23-s + (54.5 + 94.3i)25-s + (124 − 214. i)26-s + 10·29-s + ⋯ |
L(s) = 1 | + (0.707 − 1.22i)2-s + (−0.499 − 0.866i)4-s + (0.178 − 0.309i)5-s + (−0.252 − 0.438i)10-s + (0.849 + 1.47i)11-s + 1.32·13-s + (0.499 − 0.866i)16-s + (−0.599 − 1.03i)17-s + (0.603 − 1.04i)19-s − 0.357·20-s + 2.40·22-s + (−0.190 + 0.329i)23-s + (0.435 + 0.755i)25-s + (0.935 − 1.62i)26-s + 0.0640·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.400168828\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.400168828\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-2 + 3.46i)T + (-4 - 6.92i)T^{2} \) |
| 5 | \( 1 + (-2 + 3.46i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-31 - 53.6i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 62T + 2.19e3T^{2} \) |
| 17 | \( 1 + (42 + 72.7i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-50 + 86.6i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (21 - 36.3i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 10T + 2.43e4T^{2} \) |
| 31 | \( 1 + (24 + 41.5i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-123 + 213. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 248T + 6.89e4T^{2} \) |
| 43 | \( 1 - 68T + 7.95e4T^{2} \) |
| 47 | \( 1 + (162 - 280. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-129 - 223. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (60 + 103. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-311 + 538. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (452 + 782. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 678T + 3.57e5T^{2} \) |
| 73 | \( 1 + (321 + 555. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (370 - 640. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 468T + 5.71e5T^{2} \) |
| 89 | \( 1 + (100 - 173. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.26e3T + 9.12e5T^{2} \) |
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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
−10.79140244527796179132987238589, −9.533804085030187635325435411250, −9.167653644842767042435496421138, −7.58581335285119630267708267129, −6.62719138379382806629512412425, −5.17827915420782659295821722332, −4.41862198501957200140632077060, −3.39893247367747519061621449493, −2.12097258348363533111229685037, −1.07237533859292919610992333894,
1.34367975921383036261734921342, 3.40961292006127990460344089987, 4.18982102974614174638649916255, 5.67499072268838903026670057578, 6.18687543893505468745118794139, 6.86993161531219033184036837264, 8.408040075433781663016324565477, 8.483697254092750268115934490116, 10.17374828020073240013522114154, 10.97243618838578852741091962799