L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.224 + 0.389i)7-s + 0.999·8-s + (−1.72 − 2.98i)11-s + (−1.22 + 2.12i)13-s + (0.224 − 0.389i)14-s + (−0.5 − 0.866i)16-s + 5.89·17-s − 5.44·19-s + (−1.72 + 2.98i)22-s + (3.44 − 5.97i)23-s + 2.44·26-s − 0.449·28-s + (−3 − 5.19i)29-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.0849 + 0.147i)7-s + 0.353·8-s + (−0.520 − 0.900i)11-s + (−0.339 + 0.588i)13-s + (0.0600 − 0.104i)14-s + (−0.125 − 0.216i)16-s + 1.43·17-s − 1.25·19-s + (−0.367 + 0.636i)22-s + (0.719 − 1.24i)23-s + 0.480·26-s − 0.0849·28-s + (−0.557 − 0.964i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.024279925\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.024279925\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-0.224 - 0.389i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.72 + 2.98i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.22 - 2.12i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 5.89T + 17T^{2} \) |
| 19 | \( 1 + 5.44T + 19T^{2} \) |
| 23 | \( 1 + (-3.44 + 5.97i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.775 - 1.34i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.27 + 2.20i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.22 + 3.85i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 3.55T + 53T^{2} \) |
| 59 | \( 1 + (-6.62 + 11.4i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.22 + 3.85i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.27 - 3.94i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2.44T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 + (3.67 + 6.36i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2 + 3.46i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 3.10T + 89T^{2} \) |
| 97 | \( 1 + (6.5 + 11.2i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.465371871350710821992060059433, −8.467903725557168698115737347824, −8.079771546077385340243728800922, −6.97537764644931190465912176764, −6.01857063879957646581834825497, −5.04212092178310158361949310413, −4.03436958633878684338116184850, −3.00547304902650612275748777160, −2.04290339753211947038474179905, −0.51270757216369847913234352106,
1.28895602039534303826382008075, 2.70152647730484150154313557091, 3.97720622262018784983392353855, 5.05946821199069051270039225008, 5.66368460758130718116018695604, 6.74042445853328884785243528904, 7.63241776608715427184913700853, 7.933653287277455323316292036409, 9.122313407915733113654369239955, 9.748751361065571451630813336316