L(s) = 1 | + (0.623 − 0.781i)4-s + (0.826 + 0.563i)7-s + (−0.109 + 1.46i)13-s + (−0.222 − 0.974i)16-s + (0.900 − 1.56i)19-s + (0.826 − 0.563i)25-s + (0.955 − 0.294i)28-s − 1.97·31-s + (1.95 − 0.294i)37-s + (−1.21 + 1.12i)43-s + (0.365 + 0.930i)49-s + (1.07 + 0.997i)52-s + (0.455 + 0.571i)61-s + (−0.900 − 0.433i)64-s + 1.91·67-s + ⋯ |
L(s) = 1 | + (0.623 − 0.781i)4-s + (0.826 + 0.563i)7-s + (−0.109 + 1.46i)13-s + (−0.222 − 0.974i)16-s + (0.900 − 1.56i)19-s + (0.826 − 0.563i)25-s + (0.955 − 0.294i)28-s − 1.97·31-s + (1.95 − 0.294i)37-s + (−1.21 + 1.12i)43-s + (0.365 + 0.930i)49-s + (1.07 + 0.997i)52-s + (0.455 + 0.571i)61-s + (−0.900 − 0.433i)64-s + 1.91·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.712605996\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.712605996\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.826 - 0.563i)T \) |
good | 2 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 5 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 11 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 13 | \( 1 + (0.109 - 1.46i)T + (-0.988 - 0.149i)T^{2} \) |
| 17 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 19 | \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 29 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 31 | \( 1 + 1.97T + T^{2} \) |
| 37 | \( 1 + (-1.95 + 0.294i)T + (0.955 - 0.294i)T^{2} \) |
| 41 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 43 | \( 1 + (1.21 - 1.12i)T + (0.0747 - 0.997i)T^{2} \) |
| 47 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 53 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 59 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 61 | \( 1 + (-0.455 - 0.571i)T + (-0.222 + 0.974i)T^{2} \) |
| 67 | \( 1 - 1.91T + T^{2} \) |
| 71 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 73 | \( 1 + (-0.0931 - 1.24i)T + (-0.988 + 0.149i)T^{2} \) |
| 79 | \( 1 - 0.730T + T^{2} \) |
| 83 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 89 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 97 | \( 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2} \) |
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
−8.740198488078288056953886618706, −7.75818706529117394426524195583, −6.99101005944351128465239985723, −6.51521970044423044261944261599, −5.52024303783440960318875643686, −4.97889113077939177761673598026, −4.21967753728484473788519381886, −2.81250637750135099199955592724, −2.13523697470875816880840198875, −1.18454278821053759701428743417,
1.24505604772280292092283920774, 2.28355826536084198621502043544, 3.41176196922833813286496515590, 3.78086364557959086151263729998, 5.05349151886264772729134475547, 5.60198721018634062475580778178, 6.61808219788132762450352515768, 7.46867354300416737173421873066, 7.85235227210671107697029624682, 8.338385337195517509033209102626