L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.329 + 2.21i)5-s − 0.999·6-s + (0.903 + 0.521i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.820 − 2.07i)10-s + 0.877·11-s + (0.866 − 0.499i)12-s + (−3.82 − 2.20i)13-s − 1.04·14-s + (−1.39 + 1.75i)15-s + (−0.5 − 0.866i)16-s + (−4.44 + 2.56i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.147 + 0.989i)5-s − 0.408·6-s + (0.341 + 0.197i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.259 − 0.657i)10-s + 0.264·11-s + (0.249 − 0.144i)12-s + (−1.05 − 0.611i)13-s − 0.278·14-s + (−0.359 + 0.452i)15-s + (−0.125 − 0.216i)16-s + (−1.07 + 0.622i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.293i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9336770071\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9336770071\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (0.329 - 2.21i)T \) |
| 37 | \( 1 + (0.0184 - 6.08i)T \) |
good | 7 | \( 1 + (-0.903 - 0.521i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 0.877T + 11T^{2} \) |
| 13 | \( 1 + (3.82 + 2.20i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (4.44 - 2.56i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.51 - 2.62i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 7.11iT - 23T^{2} \) |
| 29 | \( 1 - 5.96T + 29T^{2} \) |
| 31 | \( 1 + 1.73T + 31T^{2} \) |
| 41 | \( 1 + (-3.12 + 5.41i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 1.46iT - 43T^{2} \) |
| 47 | \( 1 + 3.00iT - 47T^{2} \) |
| 53 | \( 1 + (11.1 - 6.44i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.32 - 7.48i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.80 + 3.13i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.30 + 1.33i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.582 + 1.00i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 5.73iT - 73T^{2} \) |
| 79 | \( 1 + (5.03 - 8.72i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (9.72 - 5.61i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.63 - 13.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 9.07iT - 97T^{2} \) |
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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
−10.13033839438143918584146605593, −9.433981741552631582270227306613, −8.471711998824937554875240387854, −7.81393851557972334240083944572, −7.06516105464813699407908618927, −6.21512989013487633148238684557, −5.16395683587291806235518460076, −3.99689361384490923505596860934, −2.87368438282599510159961857195, −1.86889664233860349970425823284,
0.44939514485174928491940230701, 1.84376136897462936538235863911, 2.77108612191599214041689570870, 4.37088121021911010221518075392, 4.72878667748577373977394929940, 6.38862346778881905346488995505, 7.17603667334308222448040994496, 8.027326112114673564753732451924, 8.787000814805981825548692003361, 9.258376662069459396232561713880