L(s) = 1 | + (0.707 + 0.707i)3-s + (−0.866 − 0.5i)5-s + (−0.965 − 0.258i)7-s + 1.00i·9-s + (−0.258 − 0.965i)15-s + (−0.500 − 0.866i)21-s + (−0.965 + 1.67i)23-s + (0.499 + 0.866i)25-s + (−0.707 + 0.707i)27-s + 1.73i·29-s + (0.707 + 0.707i)35-s + i·41-s − 0.517i·43-s + (0.500 − 0.866i)45-s + (−0.707 + 1.22i)47-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)3-s + (−0.866 − 0.5i)5-s + (−0.965 − 0.258i)7-s + 1.00i·9-s + (−0.258 − 0.965i)15-s + (−0.500 − 0.866i)21-s + (−0.965 + 1.67i)23-s + (0.499 + 0.866i)25-s + (−0.707 + 0.707i)27-s + 1.73i·29-s + (0.707 + 0.707i)35-s + i·41-s − 0.517i·43-s + (0.500 − 0.866i)45-s + (−0.707 + 1.22i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8063279850\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8063279850\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 + (0.965 + 0.258i)T \) |
good | 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - 1.73iT - T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - iT - T^{2} \) |
| 43 | \( 1 + 0.517iT - T^{2} \) |
| 47 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (1.67 - 0.965i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + 0.517T + T^{2} \) |
| 89 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.060640834086873965634731250805, −8.404653800292038519045282621695, −7.62542983614596970857715653170, −7.10124359835544428613973377741, −5.95350654470036380253677484460, −5.07535155254076354645552295172, −4.27766719934474511999680325249, −3.50958670979317578234957647762, −3.04911743384282214052360479200, −1.56159748332681379007208729406,
0.42669062067821938387986788612, 2.20379385791908296630401422755, 2.84223756950420830526606167491, 3.73616002315054518555519974145, 4.35797681119336000726478448259, 5.85626575723916940549114981350, 6.47936477786197127806356128521, 7.05273382291059803991542005594, 7.85668831776246140117252847465, 8.414942771597592131934409126378