L(s) = 1 | + (−1.54 − 2.12i)2-s + (0.535 + 1.64i)3-s + (−1.51 + 4.65i)4-s + (3.60 + 2.61i)5-s + (2.67 − 3.67i)6-s + (7.22 − 2.34i)8-s + (−2.42 + 1.76i)9-s − 11.6i·10-s + (−2.32 − 2.36i)11-s − 8.47·12-s + (2.11 + 2.91i)13-s + (−2.38 + 7.33i)15-s + (−8.22 − 5.97i)16-s + (7.49 + 2.43i)18-s + (−17.6 + 12.8i)20-s + ⋯ |
L(s) = 1 | + (−1.09 − 1.50i)2-s + (0.309 + 0.951i)3-s + (−0.756 + 2.32i)4-s + (1.61 + 1.17i)5-s + (1.09 − 1.50i)6-s + (2.55 − 0.830i)8-s + (−0.809 + 0.587i)9-s − 3.69i·10-s + (−0.700 − 0.714i)11-s − 2.44·12-s + (0.587 + 0.809i)13-s + (−0.615 + 1.89i)15-s + (−2.05 − 1.49i)16-s + (1.76 + 0.573i)18-s + (−3.94 + 2.86i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.975267 + 0.265129i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.975267 + 0.265129i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.535 - 1.64i)T \) |
| 11 | \( 1 + (2.32 + 2.36i)T \) |
| 13 | \( 1 + (-2.11 - 2.91i)T \) |
good | 2 | \( 1 + (1.54 + 2.12i)T + (-0.618 + 1.90i)T^{2} \) |
| 5 | \( 1 + (-3.60 - 2.61i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (-5.66 - 4.11i)T^{2} \) |
| 17 | \( 1 + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (9.64 - 3.13i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 0.0553iT - 43T^{2} \) |
| 47 | \( 1 + (0.816 + 2.51i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.90 + 8.94i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-8.21 + 11.3i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + (-13.0 - 9.47i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-3.26 - 4.49i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (4.66 - 6.41i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 - 4.31T + 89T^{2} \) |
| 97 | \( 1 + (-29.9 + 92.2i)T^{2} \) |
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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
−11.00956753241710731127643704670, −10.23574522813647940158258104297, −9.751758915920929355362712131228, −9.000845749688957653434743766191, −8.151713014698443222955823687903, −6.66554408622749837292453494552, −5.34212182284311481521272227570, −3.66522333865344385142514115017, −2.82522446730605939876443239820, −1.93276120662023855422499215399,
0.925836154199858621818724476920, 2.08884899655730498554734086369, 5.07088617932477524087519741090, 5.69650113364171065298145123778, 6.47599609189318813991209083722, 7.48465881295511490726373590366, 8.440481999577501422733846036022, 8.873266752318680162467484698446, 9.824892581010457317565010856830, 10.44728649518552806857773801971