| L(s) = 1 | − 1.40·2-s + 2.55·3-s − 0.0156·4-s + 1.76·5-s − 3.60·6-s − 7-s + 2.83·8-s + 3.55·9-s − 2.48·10-s − 5.40·11-s − 0.0400·12-s + 5.90·13-s + 1.40·14-s + 4.52·15-s − 3.96·16-s + 17-s − 5.00·18-s + 3.08·19-s − 0.0276·20-s − 2.55·21-s + 7.61·22-s − 6.03·23-s + 7.26·24-s − 1.87·25-s − 8.31·26-s + 1.41·27-s + 0.0156·28-s + ⋯ |
| L(s) = 1 | − 0.996·2-s + 1.47·3-s − 0.00781·4-s + 0.790·5-s − 1.47·6-s − 0.377·7-s + 1.00·8-s + 1.18·9-s − 0.786·10-s − 1.63·11-s − 0.0115·12-s + 1.63·13-s + 0.376·14-s + 1.16·15-s − 0.992·16-s + 0.242·17-s − 1.17·18-s + 0.708·19-s − 0.00617·20-s − 0.558·21-s + 1.62·22-s − 1.25·23-s + 1.48·24-s − 0.375·25-s − 1.63·26-s + 0.272·27-s + 0.00295·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9870120784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9870120784\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + 1.40T + 2T^{2} \) |
| 3 | \( 1 - 2.55T + 3T^{2} \) |
| 5 | \( 1 - 1.76T + 5T^{2} \) |
| 11 | \( 1 + 5.40T + 11T^{2} \) |
| 13 | \( 1 - 5.90T + 13T^{2} \) |
| 19 | \( 1 - 3.08T + 19T^{2} \) |
| 23 | \( 1 + 6.03T + 23T^{2} \) |
| 29 | \( 1 + 8.99T + 29T^{2} \) |
| 31 | \( 1 + 2.26T + 31T^{2} \) |
| 37 | \( 1 + 1.67T + 37T^{2} \) |
| 41 | \( 1 - 4.25T + 41T^{2} \) |
| 43 | \( 1 - 4.06T + 43T^{2} \) |
| 47 | \( 1 + 4.30T + 47T^{2} \) |
| 53 | \( 1 + 0.552T + 53T^{2} \) |
| 59 | \( 1 + 2.17T + 59T^{2} \) |
| 61 | \( 1 - 3.74T + 61T^{2} \) |
| 67 | \( 1 - 8.58T + 67T^{2} \) |
| 71 | \( 1 - 5.26T + 71T^{2} \) |
| 73 | \( 1 + 1.43T + 73T^{2} \) |
| 79 | \( 1 + 5.93T + 79T^{2} \) |
| 83 | \( 1 - 9.02T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 + 0.119T + 97T^{2} \) |
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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
−13.46687311809531244462702208145, −13.08409994157793955138210258161, −10.89194428379097658503075260062, −9.902339602993260453747427155411, −9.260976228067122008358910553751, −8.248872921460970218436831633342, −7.59902436732142338038413310247, −5.68883989278509570580671672501, −3.64621661552951217949133574307, −2.03058086985751467070216533709,
2.03058086985751467070216533709, 3.64621661552951217949133574307, 5.68883989278509570580671672501, 7.59902436732142338038413310247, 8.248872921460970218436831633342, 9.260976228067122008358910553751, 9.902339602993260453747427155411, 10.89194428379097658503075260062, 13.08409994157793955138210258161, 13.46687311809531244462702208145
