L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.258 + 0.965i)3-s + (0.500 − 0.866i)6-s + (−1.22 + 1.22i)7-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s + 1.73·14-s + 1.00·16-s + (−0.707 − 0.707i)17-s + (0.965 + 0.258i)18-s + (−1.49 − 0.866i)21-s + (−0.866 − 0.5i)24-s + (−0.707 − 0.707i)27-s + 1.00i·34-s + (0.448 + 1.67i)42-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.258 + 0.965i)3-s + (0.500 − 0.866i)6-s + (−1.22 + 1.22i)7-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s + 1.73·14-s + 1.00·16-s + (−0.707 − 0.707i)17-s + (0.965 + 0.258i)18-s + (−1.49 − 0.866i)21-s + (−0.866 − 0.5i)24-s + (−0.707 − 0.707i)27-s + 1.00i·34-s + (0.448 + 1.67i)42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.03507408012\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03507408012\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.258 - 0.965i)T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + (0.707 + 0.707i)T \) |
good | 2 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 7 | \( 1 + (1.22 - 1.22i)T - iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 59 | \( 1 + 1.73T + T^{2} \) |
| 61 | \( 1 + 2T + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - 1.73iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + 2iT - T^{2} \) |
| 83 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−8.846969264140072635669187092976, −8.179429602017002404845844061773, −6.89003689233588603859218935972, −5.99219496448879642115090565427, −5.46292908963856538781424194671, −4.56262717529636930195847749211, −3.33003097663824771469702310679, −2.80023084825287963260119746084, −2.01320590940827448571370620370, −0.02545428596973558925332537796,
1.26335389290067736961625803499, 2.75964135765979437261334562891, 3.49797638240591585257606217050, 4.31076765793652976136743215103, 5.89274220724448014794838074822, 6.45803739424543310774864200155, 6.90632470450975250288647327239, 7.63393760348121085389834375638, 8.052476914798070278362475642472, 9.090903519304288124239193319565