L(s) = 1 | + 1.36·2-s − 2.44·3-s − 0.139·4-s − 1.90·5-s − 3.33·6-s + 2.23·7-s − 2.91·8-s + 2.96·9-s − 2.59·10-s + 2.42·11-s + 0.341·12-s + 5.43·13-s + 3.04·14-s + 4.64·15-s − 3.70·16-s + 1.99·17-s + 4.03·18-s + 1.90·19-s + 0.266·20-s − 5.45·21-s + 3.30·22-s + 1.41·23-s + 7.12·24-s − 1.37·25-s + 7.41·26-s + 0.0930·27-s − 0.312·28-s + ⋯ |
L(s) = 1 | + 0.964·2-s − 1.40·3-s − 0.0699·4-s − 0.851·5-s − 1.35·6-s + 0.845·7-s − 1.03·8-s + 0.987·9-s − 0.821·10-s + 0.730·11-s + 0.0985·12-s + 1.50·13-s + 0.814·14-s + 1.20·15-s − 0.925·16-s + 0.484·17-s + 0.952·18-s + 0.438·19-s + 0.0595·20-s − 1.19·21-s + 0.704·22-s + 0.296·23-s + 1.45·24-s − 0.275·25-s + 1.45·26-s + 0.0179·27-s − 0.0590·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.264011121\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.264011121\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 547 | \( 1 - T \) |
good | 2 | \( 1 - 1.36T + 2T^{2} \) |
| 3 | \( 1 + 2.44T + 3T^{2} \) |
| 5 | \( 1 + 1.90T + 5T^{2} \) |
| 7 | \( 1 - 2.23T + 7T^{2} \) |
| 11 | \( 1 - 2.42T + 11T^{2} \) |
| 13 | \( 1 - 5.43T + 13T^{2} \) |
| 17 | \( 1 - 1.99T + 17T^{2} \) |
| 19 | \( 1 - 1.90T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 - 9.30T + 29T^{2} \) |
| 31 | \( 1 + 0.129T + 31T^{2} \) |
| 37 | \( 1 + 2.45T + 37T^{2} \) |
| 41 | \( 1 - 3.18T + 41T^{2} \) |
| 43 | \( 1 + 7.21T + 43T^{2} \) |
| 47 | \( 1 - 1.81T + 47T^{2} \) |
| 53 | \( 1 - 8.72T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 + 1.68T + 67T^{2} \) |
| 71 | \( 1 + 4.01T + 71T^{2} \) |
| 73 | \( 1 - 8.45T + 73T^{2} \) |
| 79 | \( 1 + 8.77T + 79T^{2} \) |
| 83 | \( 1 - 7.89T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−11.23311482914863290215512637463, −10.32150389386865333558127649591, −8.908270689402929447577572210830, −8.097193180281259337206966574808, −6.75201073999519846767717040691, −5.97983458205555565163638577782, −5.12316552027369861247418883060, −4.33007372888459745049990636619, −3.45638182762943610382930404966, −0.995053803263551170368980007942,
0.995053803263551170368980007942, 3.45638182762943610382930404966, 4.33007372888459745049990636619, 5.12316552027369861247418883060, 5.97983458205555565163638577782, 6.75201073999519846767717040691, 8.097193180281259337206966574808, 8.908270689402929447577572210830, 10.32150389386865333558127649591, 11.23311482914863290215512637463