| L(s) = 1 | − 1.96·2-s − 0.445·3-s + 2.85·4-s − 1.89·5-s + 0.873·6-s − 0.925·7-s − 3.63·8-s − 0.801·9-s + 3.72·10-s − 1.27·12-s − 1.34·13-s + 1.81·14-s + 0.844·15-s + 4.29·16-s − 0.690·17-s + 1.57·18-s − 1.99·19-s − 5.41·20-s + 0.411·21-s + 1.61·24-s + 2.60·25-s + 2.63·26-s + 0.801·27-s − 2.64·28-s − 1.65·30-s + 0.569·31-s − 4.78·32-s + ⋯ |
| L(s) = 1 | − 1.96·2-s − 0.445·3-s + 2.85·4-s − 1.89·5-s + 0.873·6-s − 0.925·7-s − 3.63·8-s − 0.801·9-s + 3.72·10-s − 1.27·12-s − 1.34·13-s + 1.81·14-s + 0.844·15-s + 4.29·16-s − 0.690·17-s + 1.57·18-s − 1.99·19-s − 5.41·20-s + 0.411·21-s + 1.61·24-s + 2.60·25-s + 2.63·26-s + 0.801·27-s − 2.64·28-s − 1.65·30-s + 0.569·31-s − 4.78·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.06088638697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06088638697\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1511 | \( 1 - T \) |
| good | 2 | \( 1 + 1.96T + T^{2} \) |
| 3 | \( 1 + 0.445T + T^{2} \) |
| 5 | \( 1 + 1.89T + T^{2} \) |
| 7 | \( 1 + 0.925T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.34T + T^{2} \) |
| 17 | \( 1 + 0.690T + T^{2} \) |
| 19 | \( 1 + 1.99T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 0.569T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.67T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 0.0641T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.541841320070523746680571218692, −8.702400828784885232647116797462, −8.265012116378996491938590169456, −7.44161017207300400418665433037, −6.77947256256710264119791492380, −6.16307928036549261671972007879, −4.57739388299761447182280006205, −3.26197524753029355825887968549, −2.42283524429075078657456845814, −0.31992206517266049692175942331,
0.31992206517266049692175942331, 2.42283524429075078657456845814, 3.26197524753029355825887968549, 4.57739388299761447182280006205, 6.16307928036549261671972007879, 6.77947256256710264119791492380, 7.44161017207300400418665433037, 8.265012116378996491938590169456, 8.702400828784885232647116797462, 9.541841320070523746680571218692
