L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (−0.258 + 0.965i)5-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.500i)10-s + (1.36 + 1.36i)13-s − 1.00·16-s + (−1.22 − 1.22i)17-s + (−0.965 − 0.258i)20-s + (−0.866 − 0.499i)25-s + 1.93i·26-s + 0.517·29-s + (−0.707 − 0.707i)32-s − 1.73i·34-s + (−0.366 + 0.366i)37-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (−0.258 + 0.965i)5-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.500i)10-s + (1.36 + 1.36i)13-s − 1.00·16-s + (−1.22 − 1.22i)17-s + (−0.965 − 0.258i)20-s + (−0.866 − 0.499i)25-s + 1.93i·26-s + 0.517·29-s + (−0.707 − 0.707i)32-s − 1.73i·34-s + (−0.366 + 0.366i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.466516571\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.466516571\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.258 - 0.965i)T \) |
good | 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-1.36 - 1.36i)T + iT^{2} \) |
| 17 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - 0.517T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 41 | \( 1 - 1.41iT - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.73T + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - 1.93T + T^{2} \) |
| 97 | \( 1 + (-1 + i)T - iT^{2} \) |
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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
−9.747996591352797232682278875238, −8.848385817632453182780527671625, −8.223429885019670373969688603657, −7.10951242587744901535961933587, −6.70613549832146086268397994465, −6.07877906700969303633415754468, −4.85116583072543224897640440412, −4.08560928084775992775442789835, −3.23260049113057863805176944408, −2.19390629362508981274093439500,
0.956148891287532981728575777598, 2.13995327480068837656044663596, 3.51859965101872483672066882489, 4.08679409073861427608742218626, 5.09581389565937010349993109616, 5.81783058493684125184039301571, 6.55631340333004375664980792855, 7.916908006841044297481760186517, 8.657763602076204048071120822635, 9.219236774377465034741024223827