L(s) = 1 | + (0.5 + 0.866i)7-s + 13-s + (−1.5 + 0.866i)19-s + (0.5 − 0.866i)25-s + (1.5 + 0.866i)31-s + (0.5 + 0.866i)37-s − 1.73i·43-s + (−0.499 + 0.866i)49-s + (−1 − 1.73i)61-s + (−1.5 − 0.866i)67-s + (−0.5 + 0.866i)73-s + (1.5 − 0.866i)79-s + (0.5 + 0.866i)91-s − 2·97-s + (−1.5 + 0.866i)103-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)7-s + 13-s + (−1.5 + 0.866i)19-s + (0.5 − 0.866i)25-s + (1.5 + 0.866i)31-s + (0.5 + 0.866i)37-s − 1.73i·43-s + (−0.499 + 0.866i)49-s + (−1 − 1.73i)61-s + (−1.5 − 0.866i)67-s + (−0.5 + 0.866i)73-s + (1.5 − 0.866i)79-s + (0.5 + 0.866i)91-s − 2·97-s + (−1.5 + 0.866i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.089836250\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.089836250\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
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 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + 1.73iT - T^{2} \) |
| 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 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 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
−10.42155080946776623291331768250, −9.261681444660892454132230085584, −8.405080512342872457790042388779, −8.112762697373264439384570786860, −6.60628435969078377498052152443, −6.08562677816785593985410502332, −5.00343132322739890688605824133, −4.08133127446727830685973476746, −2.82279822552738431504329786803, −1.64777075275712270920995495399,
1.24909024554417592940450729484, 2.72902753207584850539372492885, 4.05197065307616256362195188110, 4.64540335931995716799197097869, 5.94398593579786385967362859058, 6.71531333641253673567420261415, 7.65259079006108557485197483987, 8.426468370268175024764815754304, 9.207418794365086457917037546001, 10.24510717110406885069276913112