| 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
−11.32322327027195900245267695484, −10.12558584831888499888507867423, −9.214833876102256758196774750597, −8.407628981711570284840432075202, −6.96437358579711132294293366095, −6.35922327367293750340363859057, −5.07112671370704744715609432152, −4.40433009212539640800486163206, −3.28474281193956322114277116280, −1.22098439939890959777752804669,
1.91212617576908494963889095310, 3.29011427668913118006913647103, 4.35229845735795419448212843055, 5.39879003842364483761195440585, 6.07412956612036648927319428321, 7.69262683056941476863257902146, 8.401110758096744119212911979531, 9.421350655252211666791443172679, 10.34274098545387089394792086775, 11.66891425295016052225219879151
