L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)9-s + 13-s + (0.5 − 0.866i)19-s + (−0.5 − 0.866i)25-s + 0.999·27-s + (0.5 + 0.866i)31-s + (0.5 − 0.866i)37-s + (−0.5 − 0.866i)39-s + 43-s − 0.999·57-s + (1 − 1.73i)61-s + (−0.5 − 0.866i)67-s + (−0.5 − 0.866i)73-s + (−0.499 + 0.866i)75-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)9-s + 13-s + (0.5 − 0.866i)19-s + (−0.5 − 0.866i)25-s + 0.999·27-s + (0.5 + 0.866i)31-s + (0.5 − 0.866i)37-s + (−0.5 − 0.866i)39-s + 43-s − 0.999·57-s + (1 − 1.73i)61-s + (−0.5 − 0.866i)67-s + (−0.5 − 0.866i)73-s + (−0.499 + 0.866i)75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.014151757\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.014151757\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + 2T + T^{2} \) |
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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
−8.861183849603686739821246294801, −8.186813555588697684134805712111, −7.44378509326576439303765332301, −6.65035950186296513439070104807, −6.03500552218341294440449961167, −5.23411465459115133808916164584, −4.29217243239320374544618641719, −3.10690787491579656064697987943, −2.07474686449359264324201125260, −0.875647314138879257351350989160,
1.25098469745794540983897487373, 2.84695028704436942860821750156, 3.80398371324305062295748656594, 4.39180957848522038347555365599, 5.58931091868301816757841368845, 5.89148518645994271521415927600, 6.89180192732979288392052849883, 7.890531584242956379991945851906, 8.653870367119313653659351225660, 9.426353997519676712811974328904