L(s) = 1 | + (1.36 − 0.366i)2-s − 1.73·3-s + (1.73 − i)4-s + (−2.36 + 0.633i)6-s + (−3 + 3i)7-s + (1.99 − 2i)8-s + 2.99·9-s + 3.46·11-s + (−2.99 + 1.73i)12-s + (3.46 − 3.46i)13-s + (−3 + 5.19i)14-s + (1.99 − 3.46i)16-s + (4 + 4i)17-s + (4.09 − 1.09i)18-s + 3.46·19-s + ⋯ |
L(s) = 1 | + (0.965 − 0.258i)2-s − 1.00·3-s + (0.866 − 0.5i)4-s + (−0.965 + 0.258i)6-s + (−1.13 + 1.13i)7-s + (0.707 − 0.707i)8-s + 0.999·9-s + 1.04·11-s + (−0.866 + 0.499i)12-s + (0.960 − 0.960i)13-s + (−0.801 + 1.38i)14-s + (0.499 − 0.866i)16-s + (0.970 + 0.970i)17-s + (0.965 − 0.258i)18-s + 0.794·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.00572 - 0.233534i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.00572 - 0.233534i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.36 + 0.366i)T \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (3 - 3i)T - 7iT^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + (-3.46 + 3.46i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4 - 4i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 + (-1 + i)T - 23iT^{2} \) |
| 29 | \( 1 + 3.46iT - 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 37iT^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 + (1.73 - 1.73i)T - 43iT^{2} \) |
| 47 | \( 1 + (-5 - 5i)T + 47iT^{2} \) |
| 53 | \( 1 + (3.46 + 3.46i)T + 53iT^{2} \) |
| 59 | \( 1 - 10.3iT - 59T^{2} \) |
| 61 | \( 1 - 3.46iT - 61T^{2} \) |
| 67 | \( 1 + (5.19 + 5.19i)T + 67iT^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 + (6 + 6i)T + 73iT^{2} \) |
| 79 | \( 1 - 8iT - 79T^{2} \) |
| 83 | \( 1 + (-1.73 - 1.73i)T + 83iT^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + (6 - 6i)T - 97iT^{2} \) |
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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
−10.82064753659169676475932767612, −10.01251014814323967294902294909, −9.188948656186113733078742274010, −7.71106273269794550100579508414, −6.41211252149757694794069319575, −6.03937348685957357244367159652, −5.33874821107878247995170095595, −3.93668103038624653927602053104, −3.07710146786947212124114272696, −1.31094146384721399933653170648,
1.25492222982668368810838794104, 3.47064521935920326639593922001, 4.02074422806802804186106723833, 5.23068032356356192655675306017, 6.19415823766059868777666690490, 6.94182531318332556994462956122, 7.36625370848882736850121274455, 9.129928613042023674192605386036, 10.03984732236011534408164609826, 10.95717172201193101750072024972