| L(s) = 1 | − 2.28·2-s − 1.58·3-s + 3.23·4-s + 0.631·5-s + 3.62·6-s + 0.582·7-s − 2.83·8-s − 0.496·9-s − 1.44·10-s − 11-s − 5.12·12-s − 2.96·13-s − 1.33·14-s − 0.999·15-s + 0.0145·16-s + 5.10·17-s + 1.13·18-s − 0.695·19-s + 2.04·20-s − 0.921·21-s + 2.28·22-s + 5.73·23-s + 4.48·24-s − 4.60·25-s + 6.78·26-s + 5.53·27-s + 1.88·28-s + ⋯ |
| L(s) = 1 | − 1.61·2-s − 0.913·3-s + 1.61·4-s + 0.282·5-s + 1.47·6-s + 0.220·7-s − 1.00·8-s − 0.165·9-s − 0.457·10-s − 0.301·11-s − 1.47·12-s − 0.822·13-s − 0.356·14-s − 0.258·15-s + 0.00364·16-s + 1.23·17-s + 0.267·18-s − 0.159·19-s + 0.457·20-s − 0.201·21-s + 0.488·22-s + 1.19·23-s + 0.916·24-s − 0.920·25-s + 1.33·26-s + 1.06·27-s + 0.356·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 2.28T + 2T^{2} \) |
| 3 | \( 1 + 1.58T + 3T^{2} \) |
| 5 | \( 1 - 0.631T + 5T^{2} \) |
| 7 | \( 1 - 0.582T + 7T^{2} \) |
| 13 | \( 1 + 2.96T + 13T^{2} \) |
| 17 | \( 1 - 5.10T + 17T^{2} \) |
| 19 | \( 1 + 0.695T + 19T^{2} \) |
| 23 | \( 1 - 5.73T + 23T^{2} \) |
| 29 | \( 1 - 0.254T + 29T^{2} \) |
| 31 | \( 1 + 1.80T + 31T^{2} \) |
| 37 | \( 1 + 0.222T + 37T^{2} \) |
| 41 | \( 1 + 0.991T + 41T^{2} \) |
| 43 | \( 1 + 1.31T + 43T^{2} \) |
| 47 | \( 1 + 8.61T + 47T^{2} \) |
| 53 | \( 1 + 2.36T + 53T^{2} \) |
| 59 | \( 1 + 2.31T + 59T^{2} \) |
| 67 | \( 1 + 1.88T + 67T^{2} \) |
| 71 | \( 1 + 6.30T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 + 7.48T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 + 18.4T + 89T^{2} \) |
| 97 | \( 1 + 1.52T + 97T^{2} \) |
| 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
−10.02897920833360350254183581846, −9.377967186856199000930272738439, −8.361395623191973970907197074343, −7.61242157842239795884814644278, −6.75156048017796955970737443774, −5.73028205015805746939969562793, −4.84912243412886928715782536764, −2.89755663255118339419698141548, −1.46372237860879810016738663848, 0,
1.46372237860879810016738663848, 2.89755663255118339419698141548, 4.84912243412886928715782536764, 5.73028205015805746939969562793, 6.75156048017796955970737443774, 7.61242157842239795884814644278, 8.361395623191973970907197074343, 9.377967186856199000930272738439, 10.02897920833360350254183581846
