| L(s) = 1 | + (−1 + 2i)5-s − 4i·7-s − 4·11-s − 4i·17-s − 4i·23-s + (−3 − 4i)25-s − 6·29-s − 4·31-s + (8 + 4i)35-s − 8i·37-s + 10·41-s + 4i·43-s − 4i·47-s − 9·49-s − 12i·53-s + ⋯ |
| L(s) = 1 | + (−0.447 + 0.894i)5-s − 1.51i·7-s − 1.20·11-s − 0.970i·17-s − 0.834i·23-s + (−0.600 − 0.800i)25-s − 1.11·29-s − 0.718·31-s + (1.35 + 0.676i)35-s − 1.31i·37-s + 1.56·41-s + 0.609i·43-s − 0.583i·47-s − 1.28·49-s − 1.64i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.380811 - 0.616166i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.380811 - 0.616166i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1 - 2i)T \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 8iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−10.34732126367003404452157311344, −9.482518694768146588210532223282, −8.119503343433143734311525788350, −7.37999553275264887201983989654, −6.96744839759609559170902109064, −5.65796269104276392091442659837, −4.45734366663742559905138851003, −3.56793432555762107040647050628, −2.44300734529845398640788981766, −0.35992289656517384055802444759,
1.79273867690764490936497708859, 3.04865613412047538905424338313, 4.37641700718845026558430007808, 5.47446280495327654978725606812, 5.86061504132680503630264048611, 7.48504056370313229914684424488, 8.177104534353187535236921364820, 8.937430435030389903083480069754, 9.605564628161406731029364131465, 10.80129820806855198684660958484
