L(s) = 1 | + (−0.5 − 0.866i)7-s − 1.73i·13-s − 19-s − 25-s − 2·31-s + 37-s + (−0.499 + 0.866i)49-s − 1.73i·61-s − 1.73i·67-s + 1.73i·73-s − 1.73i·79-s + (−1.49 + 0.866i)91-s − 1.73i·97-s + 103-s + 2·109-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)7-s − 1.73i·13-s − 19-s − 25-s − 2·31-s + 37-s + (−0.499 + 0.866i)49-s − 1.73i·61-s − 1.73i·67-s + 1.73i·73-s − 1.73i·79-s + (−1.49 + 0.866i)91-s − 1.73i·97-s + 103-s + 2·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7667827070\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7667827070\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + 1.73iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 2T + T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.73iT - T^{2} \) |
| 67 | \( 1 + 1.73iT - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - 1.73iT - T^{2} \) |
| 79 | \( 1 + 1.73iT - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + 1.73iT - T^{2} \) |
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
−8.574731314814365310145791470677, −7.78412561309109524335628426396, −7.32970745173223716883012223910, −6.30098866653607628324910877047, −5.71898973738556147591147785606, −4.74674396255946727233850078329, −3.78775619543251877696762409812, −3.16580983972864031588914220981, −1.94461156060035397295233545608, −0.44453482659693476430196356370,
1.79145107251902452753521301044, 2.48659880096954148029614666009, 3.73762750455394345704976377007, 4.36824646562009501617559499222, 5.46230143823996702612338659992, 6.14898580901511067377874144694, 6.81736301151229187880720108905, 7.60789524748225437968582825859, 8.630524834160870791336923037494, 9.121047979608355700006923208725