| L(s) = 1 | − 0.688·2-s − 2.21·3-s − 1.52·4-s − 3.21·5-s + 1.52·6-s + 7-s + 2.42·8-s + 1.90·9-s + 2.21·10-s − 2.68·11-s + 3.37·12-s − 0.688·14-s + 7.11·15-s + 1.37·16-s + 3.59·17-s − 1.31·18-s + 8.54·19-s + 4.90·20-s − 2.21·21-s + 1.85·22-s − 3.28·23-s − 5.37·24-s + 5.33·25-s + 2.42·27-s − 1.52·28-s + 2.05·29-s − 4.90·30-s + ⋯ |
| L(s) = 1 | − 0.487·2-s − 1.27·3-s − 0.762·4-s − 1.43·5-s + 0.622·6-s + 0.377·7-s + 0.858·8-s + 0.634·9-s + 0.700·10-s − 0.810·11-s + 0.975·12-s − 0.184·14-s + 1.83·15-s + 0.344·16-s + 0.871·17-s − 0.309·18-s + 1.96·19-s + 1.09·20-s − 0.483·21-s + 0.394·22-s − 0.684·23-s − 1.09·24-s + 1.06·25-s + 0.467·27-s − 0.288·28-s + 0.380·29-s − 0.895·30-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 | \( 1 + 0.688T + 2T^{2} \) |
| 3 | \( 1 + 2.21T + 3T^{2} \) |
| 5 | \( 1 + 3.21T + 5T^{2} \) |
| 11 | \( 1 + 2.68T + 11T^{2} \) |
| 17 | \( 1 - 3.59T + 17T^{2} \) |
| 19 | \( 1 - 8.54T + 19T^{2} \) |
| 23 | \( 1 + 3.28T + 23T^{2} \) |
| 29 | \( 1 - 2.05T + 29T^{2} \) |
| 31 | \( 1 + 5.83T + 31T^{2} \) |
| 37 | \( 1 + 3.93T + 37T^{2} \) |
| 41 | \( 1 + 0.755T + 41T^{2} \) |
| 43 | \( 1 - 8.80T + 43T^{2} \) |
| 47 | \( 1 + 1.88T + 47T^{2} \) |
| 53 | \( 1 - 2.52T + 53T^{2} \) |
| 59 | \( 1 + 7.33T + 59T^{2} \) |
| 61 | \( 1 - 9.05T + 61T^{2} \) |
| 67 | \( 1 + 0.428T + 67T^{2} \) |
| 71 | \( 1 + 8.98T + 71T^{2} \) |
| 73 | \( 1 + 5.79T + 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 5.36T + 89T^{2} \) |
| 97 | \( 1 + 9.62T + 97T^{2} \) |
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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
−9.424943604970475579475895475260, −8.369901373628737910011274279396, −7.69182664678827542434551902619, −7.23095812059792818201357603362, −5.65690390623454708924253425086, −5.17505644066725341479920994358, −4.28602982382738700187627583875, −3.30005171903209262990838650036, −1.05900729119518398528258095041, 0,
1.05900729119518398528258095041, 3.30005171903209262990838650036, 4.28602982382738700187627583875, 5.17505644066725341479920994358, 5.65690390623454708924253425086, 7.23095812059792818201357603362, 7.69182664678827542434551902619, 8.369901373628737910011274279396, 9.424943604970475579475895475260
