| L(s) = 1 | + 1.13·2-s + 0.404·3-s − 0.715·4-s − 1.18·5-s + 0.458·6-s + 4.77·7-s − 3.07·8-s − 2.83·9-s − 1.34·10-s + 11-s − 0.289·12-s − 1.58·13-s + 5.41·14-s − 0.478·15-s − 2.05·16-s + 7.60·17-s − 3.21·18-s + 3.39·19-s + 0.846·20-s + 1.93·21-s + 1.13·22-s + 3.98·23-s − 1.24·24-s − 3.60·25-s − 1.79·26-s − 2.36·27-s − 3.41·28-s + ⋯ |
| L(s) = 1 | + 0.801·2-s + 0.233·3-s − 0.357·4-s − 0.529·5-s + 0.187·6-s + 1.80·7-s − 1.08·8-s − 0.945·9-s − 0.424·10-s + 0.301·11-s − 0.0836·12-s − 0.438·13-s + 1.44·14-s − 0.123·15-s − 0.514·16-s + 1.84·17-s − 0.757·18-s + 0.777·19-s + 0.189·20-s + 0.421·21-s + 0.241·22-s + 0.831·23-s − 0.254·24-s − 0.720·25-s − 0.351·26-s − 0.454·27-s − 0.646·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.481117770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.481117770\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 - 1.13T + 2T^{2} \) |
| 3 | \( 1 - 0.404T + 3T^{2} \) |
| 5 | \( 1 + 1.18T + 5T^{2} \) |
| 7 | \( 1 - 4.77T + 7T^{2} \) |
| 13 | \( 1 + 1.58T + 13T^{2} \) |
| 17 | \( 1 - 7.60T + 17T^{2} \) |
| 19 | \( 1 - 3.39T + 19T^{2} \) |
| 23 | \( 1 - 3.98T + 23T^{2} \) |
| 29 | \( 1 - 5.30T + 29T^{2} \) |
| 31 | \( 1 - 5.11T + 31T^{2} \) |
| 37 | \( 1 - 0.878T + 37T^{2} \) |
| 41 | \( 1 + 5.62T + 41T^{2} \) |
| 43 | \( 1 - 1.64T + 43T^{2} \) |
| 47 | \( 1 - 4.75T + 47T^{2} \) |
| 53 | \( 1 - 1.29T + 53T^{2} \) |
| 59 | \( 1 - 2.14T + 59T^{2} \) |
| 61 | \( 1 - 4.93T + 61T^{2} \) |
| 67 | \( 1 + 1.05T + 67T^{2} \) |
| 71 | \( 1 + 8.85T + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 + 3.46T + 97T^{2} \) |
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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
−9.425464517552384523787675436721, −8.451536188960769592312184070279, −8.098255100293803794429107562351, −7.26150989718647151593381264421, −5.81296379173106308887772890410, −5.23636624512082016663273980833, −4.55962262478878131502146003585, −3.58477869633991269515491659910, −2.69685046197024378632370297024, −1.06867821567359098578129686296,
1.06867821567359098578129686296, 2.69685046197024378632370297024, 3.58477869633991269515491659910, 4.55962262478878131502146003585, 5.23636624512082016663273980833, 5.81296379173106308887772890410, 7.26150989718647151593381264421, 8.098255100293803794429107562351, 8.451536188960769592312184070279, 9.425464517552384523787675436721
