| L(s) = 1 | − 2.82i·3-s − 2i·7-s − 5.00·9-s − 11-s − 6.82i·13-s − 1.17i·17-s − 5.65·21-s − 2.82i·23-s + 5.65i·27-s − 7.65·29-s + 2.82i·33-s − 3.65i·37-s − 19.3·39-s + 6·41-s + 6i·43-s + ⋯ |
| L(s) = 1 | − 1.63i·3-s − 0.755i·7-s − 1.66·9-s − 0.301·11-s − 1.89i·13-s − 0.284i·17-s − 1.23·21-s − 0.589i·23-s + 1.08i·27-s − 1.42·29-s + 0.492i·33-s − 0.601i·37-s − 3.09·39-s + 0.937·41-s + 0.914i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.125744829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.125744829\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 2.82iT - 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 13 | \( 1 + 6.82iT - 13T^{2} \) |
| 17 | \( 1 + 1.17iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.82iT - 23T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 3.65iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 - 0.343iT - 53T^{2} \) |
| 59 | \( 1 + 9.65T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 4.48iT - 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 6.82iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 9.31T + 89T^{2} \) |
| 97 | \( 1 - 7.65iT - 97T^{2} \) |
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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
−7.82016276272248422360238515014, −7.28271424265949400778143308535, −6.55208054822276087785797236775, −5.77276976347272379246783095079, −5.17683042252369601916232783963, −3.91665563487495587026270414267, −2.96284283410200394698271057480, −2.20936687980085743621370458852, −1.04311317275850533841363194680, −0.34400126896351502141840101761,
1.81773536746143961615545068414, 2.74893653413124035531880671984, 3.83666331154278736473582479413, 4.19730382898196538039131670779, 5.14932839455133888249026831273, 5.64182559165191288707948293515, 6.52997398904420830765205917324, 7.43739744392661635512251305607, 8.469642015956613973842884847210, 9.035781519162942642612775735177
