| L(s) = 1 | + (−3.23 − 2.35i)2-s + 3.92·3-s + (4.94 + 15.2i)4-s + (18.6 + 57.3i)5-s + (−12.6 − 9.21i)6-s + (152. − 110. i)7-s + (19.7 − 60.8i)8-s − 227.·9-s + (74.5 − 229. i)10-s + (−192. + 593. i)11-s + (19.3 + 59.6i)12-s + (−159. − 116. i)13-s − 755.·14-s + (73.1 + 225. i)15-s + (−207. + 150. i)16-s + (−471. + 1.45e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + 0.251·3-s + (0.154 + 0.475i)4-s + (0.333 + 1.02i)5-s + (−0.143 − 0.104i)6-s + (1.17 − 0.855i)7-s + (0.109 − 0.336i)8-s − 0.936·9-s + (0.235 − 0.726i)10-s + (−0.480 + 1.47i)11-s + (0.0388 + 0.119i)12-s + (−0.262 − 0.190i)13-s − 1.02·14-s + (0.0839 + 0.258i)15-s + (−0.202 + 0.146i)16-s + (−0.395 + 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.18320 + 0.726990i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18320 + 0.726990i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.23 + 2.35i)T \) |
| 41 | \( 1 + (7.38e3 + 7.83e3i)T \) |
| good | 3 | \( 1 - 3.92T + 243T^{2} \) |
| 5 | \( 1 + (-18.6 - 57.3i)T + (-2.52e3 + 1.83e3i)T^{2} \) |
| 7 | \( 1 + (-152. + 110. i)T + (5.19e3 - 1.59e4i)T^{2} \) |
| 11 | \( 1 + (192. - 593. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (159. + 116. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (471. - 1.45e3i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-1.32e3 + 962. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-2.82e3 - 2.05e3i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-935. - 2.88e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (3.23e3 - 9.95e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-1.34e3 - 4.12e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 43 | \( 1 + (-1.62e3 - 1.18e3i)T + (4.54e7 + 1.39e8i)T^{2} \) |
| 47 | \( 1 + (-1.83e4 - 1.33e4i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-1.15e3 - 3.56e3i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (3.00e4 + 2.18e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (2.47e4 - 1.79e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (7.29e3 + 2.24e4i)T + (-1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-2.15e4 + 6.62e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 - 2.89e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.09e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.31e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-6.82e4 + 4.96e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-9.08e3 - 2.79e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−13.68940183437235566319884503021, −12.27839285781168150728798172060, −10.87673361562740653963572279795, −10.58519345066864473688514794769, −9.137291689586082152509684003090, −7.78252139236622602198165133571, −6.97266277631260618337169919286, −4.93522604986421826468071288612, −3.09197001442100611386398017174, −1.68848598605328620119162604117,
0.69833252486794339421162845431, 2.49648718049880989214203125383, 5.07941369776150263970577361841, 5.77272334950412626200463186196, 7.83440374745664283254539199673, 8.676485803154654670190278840161, 9.272949986453408250775933775780, 11.10220709813644554127957013392, 11.81332772516790808145389892021, 13.42782991907127674704341070224
