| L(s) = 1 | − 2i·7-s − 4·11-s − 2i·13-s + 5i·17-s + 5·19-s − i·23-s − 2·29-s + 7·31-s + 6i·37-s + 4i·43-s + 4i·47-s + 3·49-s − 9i·53-s + 14·59-s − 11·61-s + ⋯ |
| L(s) = 1 | − 0.755i·7-s − 1.20·11-s − 0.554i·13-s + 1.21i·17-s + 1.14·19-s − 0.208i·23-s − 0.371·29-s + 1.25·31-s + 0.986i·37-s + 0.609i·43-s + 0.583i·47-s + 0.428·49-s − 1.23i·53-s + 1.82·59-s − 1.40·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.572430336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.572430336\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 5iT - 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 - 14T + 59T^{2} \) |
| 61 | \( 1 + 11T + 61T^{2} \) |
| 67 | \( 1 + 14iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 - 3T + 79T^{2} \) |
| 83 | \( 1 - iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 16iT - 97T^{2} \) |
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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
−7.975715896203517819245614427972, −7.51341788346161165351386086067, −6.62069475559202915648586693719, −5.88320244115332881198355048646, −5.11400302517237359093259604316, −4.42867012941824790532438985113, −3.45089141863871636893810066595, −2.81095020440496331154088398882, −1.62541473600245785047055149123, −0.51320670619519088892062977641,
0.902818176387185737754311669214, 2.32446047100163260834764846634, 2.75859437368975164278226958323, 3.80465457221140442853936743674, 4.84453364938573235203482031350, 5.38505375582839570961371897164, 5.97811384516288082467007100774, 7.10789473179686956302803286314, 7.44615503305948131248514102896, 8.345270183361947044599939969315
