| L(s) = 1 | + (−1.31 + 0.509i)2-s + (−0.941 − 1.45i)3-s + (1.48 − 1.34i)4-s + (4.17 − 1.24i)5-s + (1.98 + 1.43i)6-s + (3.66 − 1.58i)7-s + (−1.26 + 2.52i)8-s + (−1.22 + 2.73i)9-s + (−4.86 + 3.77i)10-s + (−0.992 + 0.235i)11-s + (−3.34 − 0.887i)12-s + (0.436 + 0.287i)13-s + (−4.02 + 3.95i)14-s + (−5.74 − 4.88i)15-s + (0.386 − 3.98i)16-s + (1.28 + 3.52i)17-s + ⋯ |
| L(s) = 1 | + (−0.932 + 0.360i)2-s + (−0.543 − 0.839i)3-s + (0.740 − 0.672i)4-s + (1.86 − 0.558i)5-s + (0.809 + 0.587i)6-s + (1.38 − 0.597i)7-s + (−0.448 + 0.893i)8-s + (−0.408 + 0.912i)9-s + (−1.53 + 1.19i)10-s + (−0.299 + 0.0709i)11-s + (−0.966 − 0.256i)12-s + (0.121 + 0.0797i)13-s + (−1.07 + 1.05i)14-s + (−1.48 − 1.26i)15-s + (0.0966 − 0.995i)16-s + (0.311 + 0.855i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02317 - 0.423147i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02317 - 0.423147i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.31 - 0.509i)T \) |
| 3 | \( 1 + (0.941 + 1.45i)T \) |
| good | 5 | \( 1 + (-4.17 + 1.24i)T + (4.17 - 2.74i)T^{2} \) |
| 7 | \( 1 + (-3.66 + 1.58i)T + (4.80 - 5.09i)T^{2} \) |
| 11 | \( 1 + (0.992 - 0.235i)T + (9.82 - 4.93i)T^{2} \) |
| 13 | \( 1 + (-0.436 - 0.287i)T + (5.14 + 11.9i)T^{2} \) |
| 17 | \( 1 + (-1.28 - 3.52i)T + (-13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (-0.0265 + 0.0729i)T + (-14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (2.50 - 5.81i)T + (-15.7 - 16.7i)T^{2} \) |
| 29 | \( 1 + (3.96 - 0.230i)T + (28.8 - 3.36i)T^{2} \) |
| 31 | \( 1 + (-0.0156 - 0.134i)T + (-30.1 + 7.14i)T^{2} \) |
| 37 | \( 1 + (6.38 + 5.35i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (1.58 + 3.16i)T + (-24.4 + 32.8i)T^{2} \) |
| 43 | \( 1 + (4.87 - 4.60i)T + (2.50 - 42.9i)T^{2} \) |
| 47 | \( 1 + (6.49 + 0.759i)T + (45.7 + 10.8i)T^{2} \) |
| 53 | \( 1 + (1.03 + 0.599i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.50 + 0.356i)T + (52.7 + 26.4i)T^{2} \) |
| 61 | \( 1 + (-3.62 + 4.86i)T + (-17.4 - 58.4i)T^{2} \) |
| 67 | \( 1 + (6.64 + 0.386i)T + (66.5 + 7.77i)T^{2} \) |
| 71 | \( 1 + (-0.674 + 3.82i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-1.85 - 10.5i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.801 + 1.59i)T + (-47.1 - 63.3i)T^{2} \) |
| 83 | \( 1 + (-9.99 - 5.01i)T + (49.5 + 66.5i)T^{2} \) |
| 89 | \( 1 + (7.23 - 1.27i)T + (83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (4.63 - 15.4i)T + (-81.0 - 53.3i)T^{2} \) |
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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
−11.19750192060726081523485137562, −10.52942142680058714830641909275, −9.660314841213899671583600064070, −8.526735416877495130952152888457, −7.77396956244041260629852215513, −6.68401409976513034144087552830, −5.65505846386365900269246610131, −5.13801938927764909374156492589, −1.94142670228250492147975013223, −1.43145665338278981053654886560,
1.75765621973823423273389578035, 2.92544435687613102922440669100, 4.94247465957082165055226895101, 5.81239533786316119571046086118, 6.81021577887245274442292359824, 8.334073356507956557044120541236, 9.179011081286126376766431991648, 10.00626375151884663728509474719, 10.59259209301323868578800748101, 11.36319680988207142938640123534
