| L(s) = 1 | + 2-s − 2.68·3-s + 4-s + 2.21·5-s − 2.68·6-s + 4.45·7-s + 8-s + 4.20·9-s + 2.21·10-s − 11-s − 2.68·12-s − 0.0499·13-s + 4.45·14-s − 5.93·15-s + 16-s + 2.30·17-s + 4.20·18-s + 3.92·19-s + 2.21·20-s − 11.9·21-s − 22-s + 2.67·23-s − 2.68·24-s − 0.106·25-s − 0.0499·26-s − 3.23·27-s + 4.45·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.54·3-s + 0.5·4-s + 0.989·5-s − 1.09·6-s + 1.68·7-s + 0.353·8-s + 1.40·9-s + 0.699·10-s − 0.301·11-s − 0.774·12-s − 0.0138·13-s + 1.19·14-s − 1.53·15-s + 0.250·16-s + 0.559·17-s + 0.991·18-s + 0.900·19-s + 0.494·20-s − 2.61·21-s − 0.213·22-s + 0.556·23-s − 0.547·24-s − 0.0212·25-s − 0.00979·26-s − 0.623·27-s + 0.842·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.934391015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.934391015\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 + 2.68T + 3T^{2} \) |
| 5 | \( 1 - 2.21T + 5T^{2} \) |
| 7 | \( 1 - 4.45T + 7T^{2} \) |
| 13 | \( 1 + 0.0499T + 13T^{2} \) |
| 17 | \( 1 - 2.30T + 17T^{2} \) |
| 19 | \( 1 - 3.92T + 19T^{2} \) |
| 23 | \( 1 - 2.67T + 23T^{2} \) |
| 29 | \( 1 + 4.38T + 29T^{2} \) |
| 31 | \( 1 - 4.46T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 - 0.808T + 41T^{2} \) |
| 43 | \( 1 + 1.84T + 43T^{2} \) |
| 47 | \( 1 + 1.34T + 47T^{2} \) |
| 53 | \( 1 - 4.19T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 + 7.78T + 61T^{2} \) |
| 67 | \( 1 - 9.32T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 7.02T + 73T^{2} \) |
| 79 | \( 1 - 4.15T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 2.44T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−8.044798814777383079100250490727, −7.53739238038149684857307240819, −6.63487998881517020088206376693, −5.76865848766953300443873019310, −5.53762708707053002521434361212, −4.84369454206577325982820029591, −4.29907083318204431178365741617, −2.85915886396175669696770926032, −1.74418780605455940949099166058, −1.04098426726392884924769590593,
1.04098426726392884924769590593, 1.74418780605455940949099166058, 2.85915886396175669696770926032, 4.29907083318204431178365741617, 4.84369454206577325982820029591, 5.53762708707053002521434361212, 5.76865848766953300443873019310, 6.63487998881517020088206376693, 7.53739238038149684857307240819, 8.044798814777383079100250490727
