L(s) = 1 | − 1.41i·2-s + (−1 − 1.41i)3-s − 2.00·4-s + 4·5-s + (−2.00 + 1.41i)6-s + 2.82i·7-s + 2.82i·8-s + (−1.00 + 2.82i)9-s − 5.65i·10-s − 5.65i·11-s + (2.00 + 2.82i)12-s − 5.65i·13-s + 4.00·14-s + (−4 − 5.65i)15-s + 4.00·16-s − 2.82i·17-s + ⋯ |
L(s) = 1 | − 0.999i·2-s + (−0.577 − 0.816i)3-s − 1.00·4-s + 1.78·5-s + (−0.816 + 0.577i)6-s + 1.06i·7-s + 1.00i·8-s + (−0.333 + 0.942i)9-s − 1.78i·10-s − 1.70i·11-s + (0.577 + 0.816i)12-s − 1.56i·13-s + 1.06·14-s + (−1.03 − 1.46i)15-s + 1.00·16-s − 0.685i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.411110 - 1.29346i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.411110 - 1.29346i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + (1 + 1.41i)T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 4T + 5T^{2} \) |
| 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 + 5.65iT - 13T^{2} \) |
| 17 | \( 1 + 2.82iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 5.65iT - 31T^{2} \) |
| 37 | \( 1 - 2.82iT - 37T^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 - 2.82iT - 59T^{2} \) |
| 61 | \( 1 + 2.82iT - 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 2.82iT - 79T^{2} \) |
| 83 | \( 1 - 5.65iT - 83T^{2} \) |
| 89 | \( 1 - 8.48iT - 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−10.54797172680633314239533396578, −9.755220705946178459626759727707, −8.754380700887461843055379533094, −8.122480679902390433930843126472, −6.31708903127929688712564443784, −5.61964960716596231119475507921, −5.27090920443695448049474379147, −2.96029861840373608926895350400, −2.29469906273664731539464218700, −0.899170683922299266067064534261,
1.73894201865689074860805050848, 4.03510961979211464119430070003, 4.70548370556210171918500755311, 5.64712345874069948308110651692, 6.69139225764459790538423052308, 7.00512274332638751610006117946, 8.775569190695483673964856257376, 9.479189093592101052369520249476, 10.12104354816596893106219656978, 10.57041398956212425591310601606