L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.866 − 0.5i)3-s − 1.00i·4-s + (−0.844 + 3.15i)5-s + (−0.258 + 0.965i)6-s + (2.31 + 1.28i)7-s + (0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (−1.63 − 2.82i)10-s + (−0.349 + 1.30i)11-s + (−0.500 − 0.866i)12-s + (3.41 − 1.16i)13-s + (−2.54 + 0.729i)14-s + (0.844 + 3.15i)15-s − 1.00·16-s − 2.35·17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (0.499 − 0.288i)3-s − 0.500i·4-s + (−0.377 + 1.40i)5-s + (−0.105 + 0.394i)6-s + (0.874 + 0.484i)7-s + (0.250 + 0.250i)8-s + (0.166 − 0.288i)9-s + (−0.515 − 0.893i)10-s + (−0.105 + 0.393i)11-s + (−0.144 − 0.250i)12-s + (0.946 − 0.322i)13-s + (−0.679 + 0.194i)14-s + (0.218 + 0.813i)15-s − 0.250·16-s − 0.570·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.171 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.825694 + 0.981597i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.825694 + 0.981597i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (-2.31 - 1.28i)T \) |
| 13 | \( 1 + (-3.41 + 1.16i)T \) |
good | 5 | \( 1 + (0.844 - 3.15i)T + (-4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (0.349 - 1.30i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + 2.35T + 17T^{2} \) |
| 19 | \( 1 + (4.46 - 1.19i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 - 0.907iT - 23T^{2} \) |
| 29 | \( 1 + (4.41 - 7.64i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.85 + 0.496i)T + (26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (-8.06 - 8.06i)T + 37iT^{2} \) |
| 41 | \( 1 + (-2.56 + 0.687i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (1.72 - 0.998i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-7.02 - 1.88i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.56 + 6.17i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.25 + 4.25i)T - 59iT^{2} \) |
| 61 | \( 1 + (-0.632 - 0.365i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (11.9 + 3.19i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (2.14 + 0.576i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (1.66 + 6.21i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (6.18 + 10.7i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.48 - 2.48i)T + 83iT^{2} \) |
| 89 | \( 1 + (3.26 - 3.26i)T - 89iT^{2} \) |
| 97 | \( 1 + (-2.33 + 8.72i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−10.94823055355137013782434543022, −10.24927942376767951801493181395, −9.016139451444390983802843899648, −8.277001646544728649056561976568, −7.53086814568012021269077071157, −6.71269546817548209560736450476, −5.85668393818340518065541161796, −4.37648667908332680686950979017, −3.03623251307203976626177136341, −1.83032902256543088398679784700,
0.878894050447220076008600061236, 2.22712566182686067218768085492, 4.11568762137921043881814017616, 4.33343638049711946279462109122, 5.79397491166971527123400439368, 7.38474191585791597793927463524, 8.324402415931564719060489602473, 8.673933486720198865255688404846, 9.463935113234111900930461430127, 10.68841325644200221158295167784