| L(s) = 1 | − 1.17·2-s + 0.539·3-s − 0.630·4-s + 0.460·5-s − 0.630·6-s − 7-s + 3.07·8-s − 2.70·9-s − 0.539·10-s + 0.829·11-s − 0.340·12-s + 1.17·14-s + 0.248·15-s − 2.34·16-s − 2.87·17-s + 3.17·18-s + 4.32·19-s − 0.290·20-s − 0.539·21-s − 0.971·22-s + 5.04·23-s + 1.65·24-s − 4.78·25-s − 3.07·27-s + 0.630·28-s + 0.261·29-s − 0.290·30-s + ⋯ |
| L(s) = 1 | − 0.827·2-s + 0.311·3-s − 0.315·4-s + 0.206·5-s − 0.257·6-s − 0.377·7-s + 1.08·8-s − 0.903·9-s − 0.170·10-s + 0.250·11-s − 0.0981·12-s + 0.312·14-s + 0.0641·15-s − 0.585·16-s − 0.698·17-s + 0.747·18-s + 0.992·19-s − 0.0650·20-s − 0.117·21-s − 0.207·22-s + 1.05·23-s + 0.338·24-s − 0.957·25-s − 0.592·27-s + 0.119·28-s + 0.0486·29-s − 0.0530·30-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
| \(L(1)\) |
\(\approx\) |
\(0.9054999761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9054999761\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.17T + 2T^{2} \) |
| 3 | \( 1 - 0.539T + 3T^{2} \) |
| 5 | \( 1 - 0.460T + 5T^{2} \) |
| 11 | \( 1 - 0.829T + 11T^{2} \) |
| 17 | \( 1 + 2.87T + 17T^{2} \) |
| 19 | \( 1 - 4.32T + 19T^{2} \) |
| 23 | \( 1 - 5.04T + 23T^{2} \) |
| 29 | \( 1 - 0.261T + 29T^{2} \) |
| 31 | \( 1 - 6.80T + 31T^{2} \) |
| 37 | \( 1 - 9.51T + 37T^{2} \) |
| 41 | \( 1 + 6.68T + 41T^{2} \) |
| 43 | \( 1 + 0.418T + 43T^{2} \) |
| 47 | \( 1 - 9.24T + 47T^{2} \) |
| 53 | \( 1 - 1.63T + 53T^{2} \) |
| 59 | \( 1 + 2.78T + 59T^{2} \) |
| 61 | \( 1 - 7.26T + 61T^{2} \) |
| 67 | \( 1 + 5.07T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 0.353T + 73T^{2} \) |
| 79 | \( 1 - 2.81T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 5.43T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−9.521769442947571207567160949336, −9.065867343530271789695503557600, −8.295026319821216538636588807231, −7.56965636087640590303727796955, −6.57440497497880168047330685009, −5.57247632240827729326083193489, −4.58819501182421502292973590935, −3.45785869643102227876086493136, −2.32912559074355418817301360524, −0.800333982960221957166513757844,
0.800333982960221957166513757844, 2.32912559074355418817301360524, 3.45785869643102227876086493136, 4.58819501182421502292973590935, 5.57247632240827729326083193489, 6.57440497497880168047330685009, 7.56965636087640590303727796955, 8.295026319821216538636588807231, 9.065867343530271789695503557600, 9.521769442947571207567160949336
