| L(s) = 1 | + i·2-s − 4-s − i·7-s − i·8-s + 3·11-s + 4i·13-s + 14-s + 16-s − 2·19-s + 3i·22-s − 6i·23-s − 4·26-s + i·28-s + 6·29-s + 5·31-s + i·32-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.377i·7-s − 0.353i·8-s + 0.904·11-s + 1.10i·13-s + 0.267·14-s + 0.250·16-s − 0.458·19-s + 0.639i·22-s − 1.25i·23-s − 0.784·26-s + 0.188i·28-s + 1.11·29-s + 0.898·31-s + 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.630312568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.630312568\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 14iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 7iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 3iT - 83T^{2} \) |
| 89 | \( 1 + 18T + 89T^{2} \) |
| 97 | \( 1 + iT - 97T^{2} \) |
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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
−9.656858818069400644858330271013, −8.734699258940243333176450954102, −8.270579265306059676889654749916, −7.00025529809323576744175469994, −6.67703276718226439866241869333, −5.78803637084768654825584545153, −4.43363282966278761210274107648, −4.17304952892473417713261866067, −2.62886704389196821115811099912, −1.07450360904072939435424155074,
0.885440186468675407454991193680, 2.21592648694694688253795068334, 3.26428706635492863075742539651, 4.13875420892722832122800955967, 5.22191986050896946834542710229, 6.00369298263222888751651844666, 7.03552846908471693451425525831, 8.087070634214029498384351773009, 8.741675086877256691838553666420, 9.579611049041271462256862232465
