L(s) = 1 | + 1.50·2-s − 1.34·3-s + 0.267·4-s + 2.54·5-s − 2.01·6-s + 0.340·7-s − 2.60·8-s − 1.20·9-s + 3.83·10-s + 11-s − 0.359·12-s + 0.512·14-s − 3.40·15-s − 4.46·16-s − 6.84·17-s − 1.81·18-s − 2.05·19-s + 0.681·20-s − 0.456·21-s + 1.50·22-s − 6.76·23-s + 3.49·24-s + 1.46·25-s + 5.63·27-s + 0.0912·28-s − 1.25·29-s − 5.13·30-s + ⋯ |
L(s) = 1 | + 1.06·2-s − 0.774·3-s + 0.133·4-s + 1.13·5-s − 0.824·6-s + 0.128·7-s − 0.922·8-s − 0.400·9-s + 1.21·10-s + 0.301·11-s − 0.103·12-s + 0.137·14-s − 0.880·15-s − 1.11·16-s − 1.65·17-s − 0.426·18-s − 0.471·19-s + 0.152·20-s − 0.0996·21-s + 0.321·22-s − 1.41·23-s + 0.713·24-s + 0.293·25-s + 1.08·27-s + 0.0172·28-s − 0.233·29-s − 0.937·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 1.50T + 2T^{2} \) |
| 3 | \( 1 + 1.34T + 3T^{2} \) |
| 5 | \( 1 - 2.54T + 5T^{2} \) |
| 7 | \( 1 - 0.340T + 7T^{2} \) |
| 17 | \( 1 + 6.84T + 17T^{2} \) |
| 19 | \( 1 + 2.05T + 19T^{2} \) |
| 23 | \( 1 + 6.76T + 23T^{2} \) |
| 29 | \( 1 + 1.25T + 29T^{2} \) |
| 31 | \( 1 + 1.07T + 31T^{2} \) |
| 37 | \( 1 - 8.12T + 37T^{2} \) |
| 41 | \( 1 - 7.18T + 41T^{2} \) |
| 43 | \( 1 + 4.43T + 43T^{2} \) |
| 47 | \( 1 + 0.783T + 47T^{2} \) |
| 53 | \( 1 + 8.45T + 53T^{2} \) |
| 59 | \( 1 - 8.65T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 + 5.40T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 + 6.66T + 73T^{2} \) |
| 79 | \( 1 + 0.0621T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 + 8.25T + 89T^{2} \) |
| 97 | \( 1 + 7.07T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.032627506801127358345498599106, −8.108245875965740662467536324664, −6.68225712942762477633790424315, −6.09622378027650206888677692146, −5.76385545095574293489291491842, −4.74304511507371211544036196817, −4.19347569221223390779311463668, −2.85405860783575210837220800789, −1.91592893633203625821377348764, 0,
1.91592893633203625821377348764, 2.85405860783575210837220800789, 4.19347569221223390779311463668, 4.74304511507371211544036196817, 5.76385545095574293489291491842, 6.09622378027650206888677692146, 6.68225712942762477633790424315, 8.108245875965740662467536324664, 9.032627506801127358345498599106