L(s) = 1 | − 4-s + 16-s + (−0.5 − 0.866i)25-s + (1 + 1.73i)37-s + (1 − 1.73i)43-s − 64-s + 2·67-s + 2·79-s + (0.5 + 0.866i)100-s + (1 − 1.73i)109-s + ⋯ |
L(s) = 1 | − 4-s + 16-s + (−0.5 − 0.866i)25-s + (1 + 1.73i)37-s + (1 − 1.73i)43-s − 64-s + 2·67-s + 2·79-s + (0.5 + 0.866i)100-s + (1 − 1.73i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9676890712\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9676890712\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 2T + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 - 2T + T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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
−8.477871022431795773556003663855, −8.122830317066194019739345565571, −7.21374550682274048826263045610, −6.30882740193674643537076218907, −5.57306651088991432208630119969, −4.77721636568213698627598579167, −4.10985503883110089462179773535, −3.29365954359094964092295553037, −2.17288818483861411014490213707, −0.791363440723084137400474316082,
0.917085110099472635406967331121, 2.24873927156594603547048436836, 3.41014954310708957042753152505, 4.09363921636143933977865234583, 4.88573270404298101204509525849, 5.63008439754334195154618621950, 6.31611146052426385825735333878, 7.43366841098501318227454996021, 7.912239764178354488690315617026, 8.730356349045030904179477848138