| L(s) = 1 | − 15.2·5-s + 16.9i·7-s − 173. i·11-s + 43.6·13-s + 69.8·17-s − 453. i·19-s − 500. i·23-s − 393.·25-s + 290.·29-s + 1.28e3i·31-s − 258. i·35-s − 437.·37-s + 893.·41-s + 895. i·43-s + 390. i·47-s + ⋯ |
| L(s) = 1 | − 0.608·5-s + 0.346i·7-s − 1.43i·11-s + 0.258·13-s + 0.241·17-s − 1.25i·19-s − 0.946i·23-s − 0.629·25-s + 0.345·29-s + 1.33i·31-s − 0.210i·35-s − 0.319·37-s + 0.531·41-s + 0.484i·43-s + 0.176i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4693987589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4693987589\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 15.2T + 625T^{2} \) |
| 7 | \( 1 - 16.9iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 173. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 43.6T + 2.85e4T^{2} \) |
| 17 | \( 1 - 69.8T + 8.35e4T^{2} \) |
| 19 | \( 1 + 453. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 500. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 290.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.28e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 437.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 893.T + 2.82e6T^{2} \) |
| 43 | \( 1 - 895. iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 390. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 2.38e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 474. iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 2.70e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 6.78e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 5.45e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 1.05e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 4.36e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 874. iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 2.99e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 6.54e3T + 8.85e7T^{2} \) |
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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
−8.645732822411615715447422174594, −8.064452056503791145122367549901, −7.04296422724493112324789182871, −6.21678678512461195283648218004, −5.36425701431740060659283739589, −4.37473018316945865519354619367, −3.37595440613262632707777368931, −2.59016844216568946397757340342, −1.05573483953771596707467613574, −0.10697798234891322160446083847,
1.27922159262965878056594757421, 2.30661838340394090511997086599, 3.73094439486844831588898673375, 4.18069558943885665058075712020, 5.31655637756921237747934227676, 6.21643812788694479301019607313, 7.41828133956458928610838380119, 7.60340265526975083486334508464, 8.638646903908254417831311790119, 9.705213817502175742787584981328
