L(s) = 1 | + 0.381·2-s − 1.85·4-s + (0.618 − 1.07i)5-s + (2.11 + 3.66i)7-s − 1.47·8-s + (0.236 − 0.408i)10-s + 3.61·11-s + (−0.690 − 1.19i)13-s + (0.809 + 1.40i)14-s + 3.14·16-s + (3.04 + 5.27i)17-s + (0.618 − 1.07i)19-s + (−1.14 + 1.98i)20-s + 1.38·22-s + (2.19 − 3.79i)23-s + ⋯ |
L(s) = 1 | + 0.270·2-s − 0.927·4-s + (0.276 − 0.478i)5-s + (0.800 + 1.38i)7-s − 0.520·8-s + (0.0746 − 0.129i)10-s + 1.09·11-s + (−0.191 − 0.331i)13-s + (0.216 + 0.374i)14-s + 0.786·16-s + (0.738 + 1.27i)17-s + (0.141 − 0.245i)19-s + (−0.256 + 0.443i)20-s + 0.294·22-s + (0.456 − 0.791i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 - 0.423i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.44208 + 0.320486i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.44208 + 0.320486i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 43 | \( 1 + (6.5 - 0.866i)T \) |
good | 2 | \( 1 - 0.381T + 2T^{2} \) |
| 5 | \( 1 + (-0.618 + 1.07i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-2.11 - 3.66i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 3.61T + 11T^{2} \) |
| 13 | \( 1 + (0.690 + 1.19i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.04 - 5.27i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.618 + 1.07i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.19 + 3.79i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.42 + 4.20i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 9.47T + 41T^{2} \) |
| 47 | \( 1 + 1.14T + 47T^{2} \) |
| 53 | \( 1 + (0.690 - 1.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 5.09T + 59T^{2} \) |
| 61 | \( 1 + (1.42 + 2.47i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.92 + 3.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.39 + 9.35i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.927 + 1.60i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.690 - 1.19i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.01 + 13.8i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.927 - 1.60i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 9.23T + 97T^{2} \) |
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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
−11.67424772199226228549382843060, −10.40553894303618823492690906554, −9.238425515540963242494942149328, −8.790394438421281215097388746165, −8.016124011734704576909192541861, −6.31475717651506146810925655415, −5.38609937878047964828311711845, −4.69882702522476547129032413739, −3.33167041264495287755344512937, −1.55474789094885735567747694563,
1.16138409765984000592841574666, 3.31530015010412508927673336024, 4.33339917858814747828843287639, 5.16876031625726942726943532333, 6.58915341542623029717931523263, 7.46024258965996841981214748531, 8.492834043760059689621063638627, 9.644763845377715479679126894531, 10.15894534483485169998372369388, 11.38681074325896555258522418680