L(s) = 1 | + (1.16 − 0.359i)2-s + (−0.425 + 0.290i)4-s + (0.478 + 0.0721i)5-s + (1.90 + 1.83i)7-s + (−1.91 + 2.39i)8-s + (0.583 − 0.0878i)10-s + (2.56 + 2.38i)11-s + (0.866 − 3.79i)13-s + (2.87 + 1.45i)14-s + (−0.987 + 2.51i)16-s + (−0.0251 + 0.335i)17-s + (3.24 + 5.62i)19-s + (−0.224 + 0.108i)20-s + (3.84 + 1.85i)22-s + (0.169 + 2.26i)23-s + ⋯ |
L(s) = 1 | + (0.823 − 0.253i)2-s + (−0.212 + 0.145i)4-s + (0.213 + 0.0322i)5-s + (0.718 + 0.695i)7-s + (−0.675 + 0.847i)8-s + (0.184 − 0.0277i)10-s + (0.774 + 0.718i)11-s + (0.240 − 1.05i)13-s + (0.768 + 0.389i)14-s + (−0.246 + 0.629i)16-s + (−0.00610 + 0.0814i)17-s + (0.744 + 1.29i)19-s + (−0.0502 + 0.0241i)20-s + (0.819 + 0.394i)22-s + (0.0353 + 0.471i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.01372 + 0.529678i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.01372 + 0.529678i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-1.90 - 1.83i)T \) |
good | 2 | \( 1 + (-1.16 + 0.359i)T + (1.65 - 1.12i)T^{2} \) |
| 5 | \( 1 + (-0.478 - 0.0721i)T + (4.77 + 1.47i)T^{2} \) |
| 11 | \( 1 + (-2.56 - 2.38i)T + (0.822 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.866 + 3.79i)T + (-11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (0.0251 - 0.335i)T + (-16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (-3.24 - 5.62i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.169 - 2.26i)T + (-22.7 + 3.42i)T^{2} \) |
| 29 | \( 1 + (-5.84 + 2.81i)T + (18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (-2.13 + 3.69i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.62 + 3.15i)T + (13.5 + 34.4i)T^{2} \) |
| 41 | \( 1 + (-2.67 + 3.35i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (5.28 + 6.63i)T + (-9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (4.71 - 1.45i)T + (38.8 - 26.4i)T^{2} \) |
| 53 | \( 1 + (7.44 - 5.07i)T + (19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (-12.3 + 1.85i)T + (56.3 - 17.3i)T^{2} \) |
| 61 | \( 1 + (7.55 + 5.15i)T + (22.2 + 56.7i)T^{2} \) |
| 67 | \( 1 + (2.35 - 4.07i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (12.8 + 6.20i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (-7.72 - 2.38i)T + (60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 + (-0.516 - 0.893i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.93 + 12.8i)T + (-74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (-1.38 + 1.28i)T + (6.65 - 88.7i)T^{2} \) |
| 97 | \( 1 + 0.104T + 97T^{2} \) |
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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
−11.66891425295016052225219879151, −10.34274098545387089394792086775, −9.421350655252211666791443172679, −8.401110758096744119212911979531, −7.69262683056941476863257902146, −6.07412956612036648927319428321, −5.39879003842364483761195440585, −4.35229845735795419448212843055, −3.29011427668913118006913647103, −1.91212617576908494963889095310,
1.22098439939890959777752804669, 3.28474281193956322114277116280, 4.40433009212539640800486163206, 5.07112671370704744715609432152, 6.35922327367293750340363859057, 6.96437358579711132294293366095, 8.407628981711570284840432075202, 9.214833876102256758196774750597, 10.12558584831888499888507867423, 11.32322327027195900245267695484