| L(s) = 1 | + 2-s − 4-s − 5-s − 3·8-s − 10-s + 5·11-s − 5·13-s − 16-s + 3·17-s + 19-s + 20-s + 5·22-s + 3·23-s − 4·25-s − 5·26-s − 29-s + 5·32-s + 3·34-s + 3·37-s + 38-s + 3·40-s − 5·41-s − 43-s − 5·44-s + 3·46-s − 4·50-s + 5·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s − 1.06·8-s − 0.316·10-s + 1.50·11-s − 1.38·13-s − 1/4·16-s + 0.727·17-s + 0.229·19-s + 0.223·20-s + 1.06·22-s + 0.625·23-s − 4/5·25-s − 0.980·26-s − 0.185·29-s + 0.883·32-s + 0.514·34-s + 0.493·37-s + 0.162·38-s + 0.474·40-s − 0.780·41-s − 0.152·43-s − 0.753·44-s + 0.442·46-s − 0.565·50-s + 0.693·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
| 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
−8.035761552438815830951640158483, −7.31768094634102573342521283311, −6.52648007231984170585797972035, −5.73411146344401015244583219499, −4.93386640312481543471658934308, −4.29849346574444026188093101269, −3.58229122042973148403581974990, −2.81914468708619706562307664333, −1.40100157092824524656446205060, 0,
1.40100157092824524656446205060, 2.81914468708619706562307664333, 3.58229122042973148403581974990, 4.29849346574444026188093101269, 4.93386640312481543471658934308, 5.73411146344401015244583219499, 6.52648007231984170585797972035, 7.31768094634102573342521283311, 8.035761552438815830951640158483
