L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.658 + 1.60i)3-s + (−0.499 + 0.866i)4-s + (−1.00 + 0.578i)5-s + (1.05 − 1.37i)6-s + (−0.296 − 2.62i)7-s + 0.999·8-s + (−2.13 + 2.11i)9-s + (1.00 + 0.578i)10-s + (−1.85 + 3.21i)11-s + (−1.71 − 0.230i)12-s + (−0.361 + 3.58i)13-s + (−2.12 + 1.57i)14-s + (−1.58 − 1.22i)15-s + (−0.5 − 0.866i)16-s + (−0.0128 + 0.0221i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.380 + 0.924i)3-s + (−0.249 + 0.433i)4-s + (−0.448 + 0.258i)5-s + (0.431 − 0.559i)6-s + (−0.111 − 0.993i)7-s + 0.353·8-s + (−0.710 + 0.703i)9-s + (0.316 + 0.182i)10-s + (−0.559 + 0.969i)11-s + (−0.495 − 0.0664i)12-s + (−0.100 + 0.994i)13-s + (−0.568 + 0.419i)14-s + (−0.409 − 0.315i)15-s + (−0.125 − 0.216i)16-s + (−0.00310 + 0.00537i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.342785 + 0.597257i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.342785 + 0.597257i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.658 - 1.60i)T \) |
| 7 | \( 1 + (0.296 + 2.62i)T \) |
| 13 | \( 1 + (0.361 - 3.58i)T \) |
good | 5 | \( 1 + (1.00 - 0.578i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.85 - 3.21i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (0.0128 - 0.0221i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.122 + 0.211i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.36 - 1.36i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 3.51iT - 29T^{2} \) |
| 31 | \( 1 + (2.24 - 3.89i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (8.29 - 4.79i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 0.471iT - 41T^{2} \) |
| 43 | \( 1 + 1.24T + 43T^{2} \) |
| 47 | \( 1 + (0.203 - 0.117i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.44 - 3.72i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.80 + 1.61i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-10.2 + 5.91i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.24 - 3.60i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 + (0.784 - 1.35i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.01 - 5.21i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 3.64iT - 83T^{2} \) |
| 89 | \( 1 + (8.11 - 4.68i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−10.97901455580463213693513202850, −10.06213218474849369243485354687, −9.701580828167552835662853767963, −8.590384047131671122540341928622, −7.66826946760732985060640309754, −6.88736220035197464551367535103, −5.09671339304679269354392206100, −4.17202560305285678981196036129, −3.45187493022084431245148337329, −2.04063017420910802729114068698,
0.40799981559219999002109066410, 2.28150512618890324366855710033, 3.51839446926176403394508944541, 5.34137253327175335991337181150, 5.92608288889356975587539516524, 7.00786736434479192521982797623, 8.117423015917802754244216515794, 8.347122024078379223201116335500, 9.251346140786536519754101391505, 10.43000769811414531324579696011