L(s) = 1 | + (0.5 + 0.866i)2-s + (−1.08 − 1.34i)3-s + (−0.499 + 0.866i)4-s + (−2.20 + 1.27i)5-s + (0.620 − 1.61i)6-s + (2.32 − 1.25i)7-s − 0.999·8-s + (−0.624 + 2.93i)9-s + (−2.20 − 1.27i)10-s + (2.05 − 3.55i)11-s + (1.71 − 0.270i)12-s + (0.691 + 3.53i)13-s + (2.25 + 1.38i)14-s + (4.11 + 1.57i)15-s + (−0.5 − 0.866i)16-s + (−2.85 + 4.93i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.629 − 0.777i)3-s + (−0.249 + 0.433i)4-s + (−0.985 + 0.568i)5-s + (0.253 − 0.660i)6-s + (0.879 − 0.475i)7-s − 0.353·8-s + (−0.208 + 0.978i)9-s + (−0.696 − 0.402i)10-s + (0.619 − 1.07i)11-s + (0.493 − 0.0781i)12-s + (0.191 + 0.981i)13-s + (0.602 + 0.370i)14-s + (1.06 + 0.407i)15-s + (−0.125 − 0.216i)16-s + (−0.691 + 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0746 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0746 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.828681 + 0.769004i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.828681 + 0.769004i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.08 + 1.34i)T \) |
| 7 | \( 1 + (-2.32 + 1.25i)T \) |
| 13 | \( 1 + (-0.691 - 3.53i)T \) |
good | 5 | \( 1 + (2.20 - 1.27i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.05 + 3.55i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (2.85 - 4.93i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.22 - 7.32i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.13 - 0.653i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6.06iT - 29T^{2} \) |
| 31 | \( 1 + (0.00131 - 0.00227i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.13 + 3.53i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2.51iT - 41T^{2} \) |
| 43 | \( 1 - 3.31T + 43T^{2} \) |
| 47 | \( 1 + (3.68 - 2.12i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.39 - 2.53i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.68 + 0.971i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.15 - 1.82i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.80 + 1.62i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6.68T + 71T^{2} \) |
| 73 | \( 1 + (1.31 - 2.28i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.07 + 12.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 11.1iT - 83T^{2} \) |
| 89 | \( 1 + (3.31 - 1.91i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 3.91T + 97T^{2} \) |
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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
−11.33391784691120373896006662019, −10.52286118936493914897766038291, −8.820881746323463381282610772853, −7.963878804618880715560692259114, −7.43140154456775202152420764744, −6.45803274775489041514386924778, −5.73514962384643384370360496814, −4.36680303402157003035701319700, −3.55580006272774337000206599632, −1.49183963508526355050175611430,
0.70538164974734687309924930323, 2.71859491867607438181932132371, 4.19425966150440283634005608803, 4.70915612715465246794144614471, 5.46452269090960116761759752896, 6.90622207962044760935972207667, 8.062097073765251217659108298578, 9.124753425629493773962984336792, 9.712426770948211539562446412219, 10.91353515653088627437346151621