L(s) = 1 | − 1.73·7-s − 1.73i·13-s − i·19-s − 25-s − 1.73i·37-s − 2i·43-s + 1.99·49-s + 1.73i·61-s + i·67-s − 73-s − 1.73·79-s + 2.99i·91-s + 97-s + 1.73·103-s + ⋯ |
L(s) = 1 | − 1.73·7-s − 1.73i·13-s − i·19-s − 25-s − 1.73i·37-s − 2i·43-s + 1.99·49-s + 1.73i·61-s + i·67-s − 73-s − 1.73·79-s + 2.99i·91-s + 97-s + 1.73·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6342435464\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6342435464\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + 1.73T + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + 1.73iT - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + iT - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + 1.73iT - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 2iT - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - 1.73iT - T^{2} \) |
| 67 | \( 1 - iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + 1.73T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.265227174105815900397456006040, −8.669301247567022146980044254525, −7.51821779879046723797360365830, −6.99326595092749575480987268386, −5.90508806644719562779067785606, −5.51353532999741307869215834485, −4.08379331213660696936032823688, −3.25591163728466483437690393791, −2.48014192037986862179197496554, −0.46338051862625487952607940924,
1.72280138499637881989110314377, 2.99981303877586771403430792425, 3.80692219246757224978804995502, 4.69007782614118770680168609932, 6.10369930287206406779904534040, 6.36512830745111480021899452796, 7.23398982359682651920682043386, 8.203069295365769715082556465148, 9.183040230168732113106534547477, 9.702820901125660573436345466720