L(s) = 1 | − 1.35·2-s + 3.39·3-s − 0.155·4-s − 0.772·5-s − 4.61·6-s + 7-s + 2.92·8-s + 8.54·9-s + 1.04·10-s − 2.11·11-s − 0.527·12-s − 1.35·14-s − 2.62·15-s − 3.66·16-s + 5.63·17-s − 11.6·18-s + 2.99·19-s + 0.119·20-s + 3.39·21-s + 2.87·22-s + 1.24·23-s + 9.94·24-s − 4.40·25-s + 18.8·27-s − 0.155·28-s − 7.96·29-s + 3.56·30-s + ⋯ |
L(s) = 1 | − 0.960·2-s + 1.96·3-s − 0.0776·4-s − 0.345·5-s − 1.88·6-s + 0.377·7-s + 1.03·8-s + 2.84·9-s + 0.331·10-s − 0.638·11-s − 0.152·12-s − 0.362·14-s − 0.677·15-s − 0.916·16-s + 1.36·17-s − 2.73·18-s + 0.688·19-s + 0.0268·20-s + 0.741·21-s + 0.612·22-s + 0.258·23-s + 2.03·24-s − 0.880·25-s + 3.62·27-s − 0.0293·28-s − 1.47·29-s + 0.650·30-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
\(L(1)\) |
\(\approx\) |
\(1.901118807\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.901118807\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.35T + 2T^{2} \) |
| 3 | \( 1 - 3.39T + 3T^{2} \) |
| 5 | \( 1 + 0.772T + 5T^{2} \) |
| 11 | \( 1 + 2.11T + 11T^{2} \) |
| 17 | \( 1 - 5.63T + 17T^{2} \) |
| 19 | \( 1 - 2.99T + 19T^{2} \) |
| 23 | \( 1 - 1.24T + 23T^{2} \) |
| 29 | \( 1 + 7.96T + 29T^{2} \) |
| 31 | \( 1 - 1.26T + 31T^{2} \) |
| 37 | \( 1 + 4.54T + 37T^{2} \) |
| 41 | \( 1 - 9.82T + 41T^{2} \) |
| 43 | \( 1 - 2.64T + 43T^{2} \) |
| 47 | \( 1 - 7.68T + 47T^{2} \) |
| 53 | \( 1 + 0.350T + 53T^{2} \) |
| 59 | \( 1 + 5.00T + 59T^{2} \) |
| 61 | \( 1 - 4.24T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 - 8.99T + 73T^{2} \) |
| 79 | \( 1 + 6.30T + 79T^{2} \) |
| 83 | \( 1 - 2.92T + 83T^{2} \) |
| 89 | \( 1 + 1.97T + 89T^{2} \) |
| 97 | \( 1 - 7.88T + 97T^{2} \) |
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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
−9.472383286943046092783774282206, −9.011639069426055083315493694338, −8.029787829080322750536234341403, −7.75231746328340505450154821100, −7.23826948243562887842797351121, −5.37832318857272289885173091574, −4.22019023286483094146177473575, −3.46755824899646798917931106334, −2.32371169146141955987718912529, −1.22282062218325438912849796505,
1.22282062218325438912849796505, 2.32371169146141955987718912529, 3.46755824899646798917931106334, 4.22019023286483094146177473575, 5.37832318857272289885173091574, 7.23826948243562887842797351121, 7.75231746328340505450154821100, 8.029787829080322750536234341403, 9.011639069426055083315493694338, 9.472383286943046092783774282206