| L(s) = 1 | − i·3-s + i·5-s − 7-s + 2·9-s + i·11-s − 2i·13-s + 15-s − 5·17-s + 7i·19-s + i·21-s − 4·23-s − 25-s − 5i·27-s − 9i·29-s − 3·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.447i·5-s − 0.377·7-s + 0.666·9-s + 0.301i·11-s − 0.554i·13-s + 0.258·15-s − 1.21·17-s + 1.60i·19-s + 0.218i·21-s − 0.834·23-s − 0.200·25-s − 0.962i·27-s − 1.67i·29-s − 0.538·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8639888510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8639888510\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 11 | \( 1 - iT \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 19 | \( 1 - 7iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 9iT - 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 11iT - 53T^{2} \) |
| 59 | \( 1 - 8iT - 59T^{2} \) |
| 61 | \( 1 + 11iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 9T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−7.971065543905584193082429830792, −7.67919390400803111350458012055, −6.73912919429095364207137622422, −6.24082036906199960826682888216, −5.45569258845616704716244610822, −4.20705385962051293850469060709, −3.72423343997630033183160616747, −2.40872265563781809707372668528, −1.78064409613165493465073856213, −0.25671518972386998128424779180,
1.27137453737700633790644149583, 2.45138884157765580290353101560, 3.49030198720235033091067828849, 4.40940033908244642552898575217, 4.79783735234586754819018714067, 5.79316195407478990708788798025, 6.80921259054860768418784390723, 7.12290139464026684137372197799, 8.334114936101499731175666578101, 9.015536804932563529713174994246
