| L(s) = 1 | − 0.732·2-s − 1.46·4-s − 5-s − 1.73·7-s + 2.53·8-s + 0.732·10-s + 2·11-s + 1.26·14-s + 1.07·16-s − 0.732·17-s − 0.535·19-s + 1.46·20-s − 1.46·22-s − 2·23-s + 25-s + 2.53·28-s + 3.26·29-s + 4.46·31-s − 5.85·32-s + 0.535·34-s + 1.73·35-s − 10.3·37-s + 0.392·38-s − 2.53·40-s + 10.7·41-s − 9.19·43-s − 2.92·44-s + ⋯ |
| L(s) = 1 | − 0.517·2-s − 0.732·4-s − 0.447·5-s − 0.654·7-s + 0.896·8-s + 0.231·10-s + 0.603·11-s + 0.338·14-s + 0.267·16-s − 0.177·17-s − 0.122·19-s + 0.327·20-s − 0.312·22-s − 0.417·23-s + 0.200·25-s + 0.479·28-s + 0.606·29-s + 0.801·31-s − 1.03·32-s + 0.0919·34-s + 0.292·35-s − 1.70·37-s + 0.0636·38-s − 0.400·40-s + 1.67·41-s − 1.40·43-s − 0.441·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7675037516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7675037516\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.732T + 2T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 + 0.732T + 17T^{2} \) |
| 19 | \( 1 + 0.535T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 - 3.26T + 29T^{2} \) |
| 31 | \( 1 - 4.46T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 9.19T + 43T^{2} \) |
| 47 | \( 1 + 0.196T + 47T^{2} \) |
| 53 | \( 1 - 9.46T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 5.39T + 61T^{2} \) |
| 67 | \( 1 + 9.19T + 67T^{2} \) |
| 71 | \( 1 - 4.73T + 71T^{2} \) |
| 73 | \( 1 - 1.73T + 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 + 2.92T + 83T^{2} \) |
| 89 | \( 1 + 5.26T + 89T^{2} \) |
| 97 | \( 1 - 5.19T + 97T^{2} \) |
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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
−8.179933590892740647643292013156, −7.13474013500674704925053631668, −6.72866102043841466877540358048, −5.79126219530938639424521281527, −5.00217533469635484607043850408, −4.18081459508641809100852083831, −3.70750260014676454873049884269, −2.73614130213191749009157823214, −1.51705810448746013541068110101, −0.50089326723643137621685565897,
0.50089326723643137621685565897, 1.51705810448746013541068110101, 2.73614130213191749009157823214, 3.70750260014676454873049884269, 4.18081459508641809100852083831, 5.00217533469635484607043850408, 5.79126219530938639424521281527, 6.72866102043841466877540358048, 7.13474013500674704925053631668, 8.179933590892740647643292013156
