| L(s) = 1 | − 2-s + 3-s + 4-s − 1.40·5-s − 6-s + 0.699·7-s − 8-s + 9-s + 1.40·10-s − 0.704·11-s + 12-s + 5.21·13-s − 0.699·14-s − 1.40·15-s + 16-s − 17-s − 18-s + 2.27·19-s − 1.40·20-s + 0.699·21-s + 0.704·22-s − 1.10·23-s − 24-s − 3.03·25-s − 5.21·26-s + 27-s + 0.699·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.627·5-s − 0.408·6-s + 0.264·7-s − 0.353·8-s + 0.333·9-s + 0.443·10-s − 0.212·11-s + 0.288·12-s + 1.44·13-s − 0.186·14-s − 0.362·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s + 0.522·19-s − 0.313·20-s + 0.152·21-s + 0.150·22-s − 0.231·23-s − 0.204·24-s − 0.606·25-s − 1.02·26-s + 0.192·27-s + 0.132·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 + 1.40T + 5T^{2} \) |
| 7 | \( 1 - 0.699T + 7T^{2} \) |
| 11 | \( 1 + 0.704T + 11T^{2} \) |
| 13 | \( 1 - 5.21T + 13T^{2} \) |
| 19 | \( 1 - 2.27T + 19T^{2} \) |
| 23 | \( 1 + 1.10T + 23T^{2} \) |
| 29 | \( 1 + 3.75T + 29T^{2} \) |
| 31 | \( 1 + 8.19T + 31T^{2} \) |
| 37 | \( 1 + 6.51T + 37T^{2} \) |
| 41 | \( 1 + 9.74T + 41T^{2} \) |
| 43 | \( 1 - 8.97T + 43T^{2} \) |
| 47 | \( 1 + 2.70T + 47T^{2} \) |
| 53 | \( 1 + 7.34T + 53T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 3.68T + 67T^{2} \) |
| 71 | \( 1 + 0.626T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 + 2.76T + 83T^{2} \) |
| 89 | \( 1 - 9.15T + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
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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
−7.899378570750214401449250507756, −7.29299487014956208926758090640, −6.48306764686631066637186420931, −5.70066403799751396167092700065, −4.78443183863738272440509142003, −3.61776720396693652078595996922, −3.45721639861950294775943473773, −2.10087541831506929578876202436, −1.38412004754322007878109040122, 0,
1.38412004754322007878109040122, 2.10087541831506929578876202436, 3.45721639861950294775943473773, 3.61776720396693652078595996922, 4.78443183863738272440509142003, 5.70066403799751396167092700065, 6.48306764686631066637186420931, 7.29299487014956208926758090640, 7.899378570750214401449250507756
