| L(s) = 1 | − 1.79·2-s − 3-s + 1.23·4-s − 2.60·5-s + 1.79·6-s + 1.37·7-s + 1.37·8-s + 9-s + 4.69·10-s − 5.21·11-s − 1.23·12-s − 0.609·13-s − 2.46·14-s + 2.60·15-s − 4.94·16-s + 0.762·17-s − 1.79·18-s + 5.43·19-s − 3.22·20-s − 1.37·21-s + 9.38·22-s + 23-s − 1.37·24-s + 1.79·25-s + 1.09·26-s − 27-s + 1.69·28-s + ⋯ |
| L(s) = 1 | − 1.27·2-s − 0.577·3-s + 0.618·4-s − 1.16·5-s + 0.734·6-s + 0.517·7-s + 0.485·8-s + 0.333·9-s + 1.48·10-s − 1.57·11-s − 0.357·12-s − 0.169·13-s − 0.658·14-s + 0.673·15-s − 1.23·16-s + 0.185·17-s − 0.424·18-s + 1.24·19-s − 0.721·20-s − 0.299·21-s + 2.00·22-s + 0.208·23-s − 0.280·24-s + 0.359·25-s + 0.215·26-s − 0.192·27-s + 0.320·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.79T + 2T^{2} \) |
| 5 | \( 1 + 2.60T + 5T^{2} \) |
| 7 | \( 1 - 1.37T + 7T^{2} \) |
| 11 | \( 1 + 5.21T + 11T^{2} \) |
| 13 | \( 1 + 0.609T + 13T^{2} \) |
| 17 | \( 1 - 0.762T + 17T^{2} \) |
| 19 | \( 1 - 5.43T + 19T^{2} \) |
| 31 | \( 1 - 3.96T + 31T^{2} \) |
| 37 | \( 1 + 3.31T + 37T^{2} \) |
| 41 | \( 1 + 1.10T + 41T^{2} \) |
| 43 | \( 1 - 7.23T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 - 4.97T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 - 2.20T + 71T^{2} \) |
| 73 | \( 1 - 3.58T + 73T^{2} \) |
| 79 | \( 1 + 7.27T + 79T^{2} \) |
| 83 | \( 1 + 3.93T + 83T^{2} \) |
| 89 | \( 1 + 1.67T + 89T^{2} \) |
| 97 | \( 1 + 18.5T + 97T^{2} \) |
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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
−8.417317250116713890052999908227, −8.151157829046178399924579658869, −7.45033836316792452080258138168, −6.87447786527642087161949107685, −5.37243526086060586329972576392, −4.88901303105669850909543833790, −3.80454588052404640328422686168, −2.51223007709982505580447404393, −1.06224578939487790039405811184, 0,
1.06224578939487790039405811184, 2.51223007709982505580447404393, 3.80454588052404640328422686168, 4.88901303105669850909543833790, 5.37243526086060586329972576392, 6.87447786527642087161949107685, 7.45033836316792452080258138168, 8.151157829046178399924579658869, 8.417317250116713890052999908227
