| L(s) = 1 | + 23·3-s + 25·5-s + 211·7-s + 286·9-s + 121·11-s + 26·13-s + 575·15-s − 407·17-s + 1.78e3·19-s + 4.85e3·21-s − 6·23-s + 625·25-s + 989·27-s − 2.38e3·29-s + 9.45e3·31-s + 2.78e3·33-s + 5.27e3·35-s − 6.91e3·37-s + 598·39-s − 9.77e3·41-s + 3.10e3·43-s + 7.15e3·45-s − 1.42e4·47-s + 2.77e4·49-s − 9.36e3·51-s − 1.86e4·53-s + 3.02e3·55-s + ⋯ |
| L(s) = 1 | + 1.47·3-s + 0.447·5-s + 1.62·7-s + 1.17·9-s + 0.301·11-s + 0.0426·13-s + 0.659·15-s − 0.341·17-s + 1.13·19-s + 2.40·21-s − 0.00236·23-s + 1/5·25-s + 0.261·27-s − 0.527·29-s + 1.76·31-s + 0.444·33-s + 0.727·35-s − 0.830·37-s + 0.0629·39-s − 0.908·41-s + 0.256·43-s + 0.526·45-s − 0.943·47-s + 1.64·49-s − 0.503·51-s − 0.912·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.311537258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.311537258\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p^{2} T \) |
| 11 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 - 23 T + p^{5} T^{2} \) |
| 7 | \( 1 - 211 T + p^{5} T^{2} \) |
| 13 | \( 1 - 2 p T + p^{5} T^{2} \) |
| 17 | \( 1 + 407 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1789 T + p^{5} T^{2} \) |
| 23 | \( 1 + 6 T + p^{5} T^{2} \) |
| 29 | \( 1 + 2387 T + p^{5} T^{2} \) |
| 31 | \( 1 - 9453 T + p^{5} T^{2} \) |
| 37 | \( 1 + 6917 T + p^{5} T^{2} \) |
| 41 | \( 1 + 9774 T + p^{5} T^{2} \) |
| 43 | \( 1 - 3108 T + p^{5} T^{2} \) |
| 47 | \( 1 + 14290 T + p^{5} T^{2} \) |
| 53 | \( 1 + 18665 T + p^{5} T^{2} \) |
| 59 | \( 1 - 36646 T + p^{5} T^{2} \) |
| 61 | \( 1 + 22945 T + p^{5} T^{2} \) |
| 67 | \( 1 - 35848 T + p^{5} T^{2} \) |
| 71 | \( 1 - 16647 T + p^{5} T^{2} \) |
| 73 | \( 1 + 34642 T + p^{5} T^{2} \) |
| 79 | \( 1 - 20554 T + p^{5} T^{2} \) |
| 83 | \( 1 - 7674 T + p^{5} T^{2} \) |
| 89 | \( 1 + 1111 p T + p^{5} T^{2} \) |
| 97 | \( 1 - 55764 T + p^{5} T^{2} \) |
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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
−10.09405335402369317229011225089, −9.264345873612174735494058124536, −8.397898461653583449554085284599, −7.913183398574320457725525101521, −6.86484878389279693177263196831, −5.36703887888344359758437728356, −4.41194922639604512638898655282, −3.20771176597360810875101133196, −2.10139879047337931195073853998, −1.28486874002085073181358472049,
1.28486874002085073181358472049, 2.10139879047337931195073853998, 3.20771176597360810875101133196, 4.41194922639604512638898655282, 5.36703887888344359758437728356, 6.86484878389279693177263196831, 7.913183398574320457725525101521, 8.397898461653583449554085284599, 9.264345873612174735494058124536, 10.09405335402369317229011225089
