| L(s) = 1 | + 5i·3-s + 8.66i·5-s + 17.3·7-s + 2·9-s + 11i·11-s + 34.6i·13-s − 43.3·15-s + 30·17-s + 44i·19-s + 86.6i·21-s − 60.6·23-s + 49.9·25-s + 145i·27-s + 34.6i·29-s + 337.·31-s + ⋯ |
| L(s) = 1 | + 0.962i·3-s + 0.774i·5-s + 0.935·7-s + 0.0740·9-s + 0.301i·11-s + 0.739i·13-s − 0.745·15-s + 0.428·17-s + 0.531i·19-s + 0.899i·21-s − 0.549·23-s + 0.399·25-s + 1.03i·27-s + 0.221i·29-s + 1.95·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.319458880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.319458880\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - 11iT \) |
| good | 3 | \( 1 - 5iT - 27T^{2} \) |
| 5 | \( 1 - 8.66iT - 125T^{2} \) |
| 7 | \( 1 - 17.3T + 343T^{2} \) |
| 13 | \( 1 - 34.6iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 30T + 4.91e3T^{2} \) |
| 19 | \( 1 - 44iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 60.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 34.6iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 337.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 303. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 72T + 6.89e4T^{2} \) |
| 43 | \( 1 - 250iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 121.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 138. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 471iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 415. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 205iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 233.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 520T + 3.89e5T^{2} \) |
| 79 | \( 1 + 883.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 210iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 849T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.05e3T + 9.12e5T^{2} \) |
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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
−10.32077047642306693556850313816, −9.733730137503802078821977639506, −8.739556540503984978422501710491, −7.76217706235860506119301706391, −6.91252542619503893514282113657, −5.81306616124004164512280487709, −4.66765353071010870626341136554, −4.08170807223159399641069315795, −2.83207726783235087315885153623, −1.49578802282915530599257520337,
0.70668843779598047009329864279, 1.47345814279544815612366400009, 2.76576887674063802134379289970, 4.34936799668946925723373532695, 5.16542999326337121231768250689, 6.20149153645972504331684733574, 7.19750029662860690218280085003, 8.179115498051224370069299996268, 8.405400611453201882048113794445, 9.731392625603826377879158851510
