L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.224 − 1.71i)3-s + (−0.499 + 0.866i)4-s + (−0.866 + 0.5i)5-s + (1.37 − 1.05i)6-s − 1.74·7-s − 0.999·8-s + (−2.89 + 0.772i)9-s + (−0.866 − 0.499i)10-s + 4.48i·11-s + (1.59 + 0.663i)12-s + (3.14 + 1.81i)13-s + (−0.871 − 1.51i)14-s + (1.05 + 1.37i)15-s + (−0.5 − 0.866i)16-s + (−5.72 + 3.30i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.129 − 0.991i)3-s + (−0.249 + 0.433i)4-s + (−0.387 + 0.223i)5-s + (0.561 − 0.430i)6-s − 0.659·7-s − 0.353·8-s + (−0.966 + 0.257i)9-s + (−0.273 − 0.158i)10-s + 1.35i·11-s + (0.461 + 0.191i)12-s + (0.872 + 0.503i)13-s + (−0.232 − 0.403i)14-s + (0.271 + 0.354i)15-s + (−0.125 − 0.216i)16-s + (−1.38 + 0.800i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.517 - 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.517 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.448272 + 0.795207i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.448272 + 0.795207i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.224 + 1.71i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 19 | \( 1 + (-3.74 - 2.22i)T \) |
good | 7 | \( 1 + 1.74T + 7T^{2} \) |
| 11 | \( 1 - 4.48iT - 11T^{2} \) |
| 13 | \( 1 + (-3.14 - 1.81i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (5.72 - 3.30i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (1.69 + 0.977i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.271 + 0.469i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 8.66iT - 31T^{2} \) |
| 37 | \( 1 + 0.251iT - 37T^{2} \) |
| 41 | \( 1 + (2.28 + 3.95i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.13 + 3.70i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.28 + 0.743i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.0649 + 0.112i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.22 - 5.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.04 + 1.80i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.40 + 4.27i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.78 - 8.29i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.51 + 6.08i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.04 - 2.91i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 10.2iT - 83T^{2} \) |
| 89 | \( 1 + (-2.10 + 3.64i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.01 + 5.20i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.24765043097199211341605768140, −10.18113989326231701696979538938, −8.954396232643499755450538603844, −8.227000637424844273596680927750, −7.08656449501369545991732862901, −6.75468197181785663548254397809, −5.80943453027692090662805688470, −4.51933160408621925210493697613, −3.37517364952518106821149838663, −1.87933457747757673227949067939,
0.46070637837515691766471314476, 2.88826655415992120042533464423, 3.59251706540123816611377621440, 4.61350735865494528831775011820, 5.64847187819208937978649202209, 6.45182700041129172039565538986, 8.091580513532495284449862165362, 8.987944435080957307439586647883, 9.592323135205492594489375979197, 10.62450520122104583112333052846