| L(s) = 1 | − 0.381·2-s − 0.381·3-s − 1.85·4-s − 0.381·5-s + 0.145·6-s − 7-s + 1.47·8-s − 2.85·9-s + 0.145·10-s + 4.85·11-s + 0.708·12-s + 0.381·14-s + 0.145·15-s + 3.14·16-s + 7.47·17-s + 1.09·18-s − 4.85·19-s + 0.708·20-s + 0.381·21-s − 1.85·22-s + 4.47·23-s − 0.562·24-s − 4.85·25-s + 2.23·27-s + 1.85·28-s − 4.09·29-s − 0.0557·30-s + ⋯ |
| L(s) = 1 | − 0.270·2-s − 0.220·3-s − 0.927·4-s − 0.170·5-s + 0.0595·6-s − 0.377·7-s + 0.520·8-s − 0.951·9-s + 0.0461·10-s + 1.46·11-s + 0.204·12-s + 0.102·14-s + 0.0376·15-s + 0.786·16-s + 1.81·17-s + 0.256·18-s − 1.11·19-s + 0.158·20-s + 0.0833·21-s − 0.395·22-s + 0.932·23-s − 0.114·24-s − 0.970·25-s + 0.430·27-s + 0.350·28-s − 0.759·29-s − 0.0101·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.381T + 2T^{2} \) |
| 3 | \( 1 + 0.381T + 3T^{2} \) |
| 5 | \( 1 + 0.381T + 5T^{2} \) |
| 11 | \( 1 - 4.85T + 11T^{2} \) |
| 17 | \( 1 - 7.47T + 17T^{2} \) |
| 19 | \( 1 + 4.85T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 + 4.09T + 29T^{2} \) |
| 31 | \( 1 + 8.70T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 5.23T + 41T^{2} \) |
| 43 | \( 1 + 7.56T + 43T^{2} \) |
| 47 | \( 1 + 2.23T + 47T^{2} \) |
| 53 | \( 1 - 8.23T + 53T^{2} \) |
| 59 | \( 1 + 2.23T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 0.708T + 67T^{2} \) |
| 71 | \( 1 + 8.18T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 6.70T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−9.242418006721246493072843423992, −8.725758535407408592816681592195, −7.87717089995583770443176939577, −6.88671388003764420764706554300, −5.85852497429691635704084424954, −5.19769500022047440090298103222, −3.92723195464011465965801762874, −3.35764240683341862665267275000, −1.48942761865667432654331216995, 0,
1.48942761865667432654331216995, 3.35764240683341862665267275000, 3.92723195464011465965801762874, 5.19769500022047440090298103222, 5.85852497429691635704084424954, 6.88671388003764420764706554300, 7.87717089995583770443176939577, 8.725758535407408592816681592195, 9.242418006721246493072843423992
