L(s) = 1 | + (0.806 + 1.16i)2-s + (−0.699 + 1.87i)4-s − 0.746·7-s + (−2.74 + 0.699i)8-s − 5.36i·11-s − 2.92i·13-s + (−0.601 − 0.866i)14-s + (−3.02 − 2.62i)16-s + 2.13·17-s + 1.73i·19-s + (6.22 − 4.32i)22-s + 7.49·23-s + (3.39 − 2.35i)26-s + (0.521 − 1.39i)28-s − 6.74i·29-s + ⋯ |
L(s) = 1 | + (0.570 + 0.821i)2-s + (−0.349 + 0.936i)4-s − 0.282·7-s + (−0.968 + 0.247i)8-s − 1.61i·11-s − 0.811i·13-s + (−0.160 − 0.231i)14-s + (−0.755 − 0.655i)16-s + 0.517·17-s + 0.397i·19-s + (1.32 − 0.921i)22-s + 1.56·23-s + (0.666 − 0.462i)26-s + (0.0985 − 0.264i)28-s − 1.25i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.998695977\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.998695977\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.806 - 1.16i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 0.746T + 7T^{2} \) |
| 11 | \( 1 + 5.36iT - 11T^{2} \) |
| 13 | \( 1 + 2.92iT - 13T^{2} \) |
| 17 | \( 1 - 2.13T + 17T^{2} \) |
| 19 | \( 1 - 1.73iT - 19T^{2} \) |
| 23 | \( 1 - 7.49T + 23T^{2} \) |
| 29 | \( 1 + 6.74iT - 29T^{2} \) |
| 31 | \( 1 - 2.64T + 31T^{2} \) |
| 37 | \( 1 - 1.07iT - 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 - 7.44iT - 43T^{2} \) |
| 47 | \( 1 - 1.73T + 47T^{2} \) |
| 53 | \( 1 + 7.72iT - 53T^{2} \) |
| 59 | \( 1 + 6.85iT - 59T^{2} \) |
| 61 | \( 1 + 6.45iT - 61T^{2} \) |
| 67 | \( 1 - 7.44iT - 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 0.690T + 73T^{2} \) |
| 79 | \( 1 - 2.64T + 79T^{2} \) |
| 83 | \( 1 - 5.85iT - 83T^{2} \) |
| 89 | \( 1 + 7.59T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−9.115726452449161618122028401331, −8.221202035198546409861692705750, −7.85189956077013771129012703076, −6.76866788729576619167533833111, −5.98493182464189907680103581857, −5.49384048734057389665599177302, −4.45533169120703348974449859219, −3.36272665139321949433189804206, −2.86014267371486368250676389943, −0.69849182332514504161715399679,
1.22572114099156985419083186889, 2.30787486215559479927093044459, 3.23583798925990949942502778059, 4.36412849263797101292094622818, 4.85487528008498155119689290777, 5.86076670221262715971274276672, 6.88773199319206074744192124046, 7.39389428353887153187992289728, 8.955757051518340124362863211992, 9.251956166203839980078685735694