L(s) = 1 | + 0.414i·3-s − 4.82i·7-s + 2.82·9-s − 3.24·11-s + 5.65i·13-s − 5.82i·17-s − 4.41·19-s + 1.99·21-s − 0.828i·23-s + 2.41i·27-s + 8·29-s + 4.82·31-s − 1.34i·33-s − 3.65i·37-s − 2.34·39-s + ⋯ |
L(s) = 1 | + 0.239i·3-s − 1.82i·7-s + 0.942·9-s − 0.977·11-s + 1.56i·13-s − 1.41i·17-s − 1.01·19-s + 0.436·21-s − 0.172i·23-s + 0.464i·27-s + 1.48·29-s + 0.867·31-s − 0.233i·33-s − 0.601i·37-s − 0.375·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.255545437\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.255545437\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 0.414iT - 3T^{2} \) |
| 7 | \( 1 + 4.82iT - 7T^{2} \) |
| 11 | \( 1 + 3.24T + 11T^{2} \) |
| 13 | \( 1 - 5.65iT - 13T^{2} \) |
| 17 | \( 1 + 5.82iT - 17T^{2} \) |
| 19 | \( 1 + 4.41T + 19T^{2} \) |
| 23 | \( 1 + 0.828iT - 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - 4.82T + 31T^{2} \) |
| 37 | \( 1 + 3.65iT - 37T^{2} \) |
| 41 | \( 1 + 0.656T + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 - 1.65iT - 47T^{2} \) |
| 53 | \( 1 + 3.65iT - 53T^{2} \) |
| 59 | \( 1 + 7.65T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 11.2iT - 67T^{2} \) |
| 71 | \( 1 - 1.65T + 71T^{2} \) |
| 73 | \( 1 + 0.171iT - 73T^{2} \) |
| 79 | \( 1 + 7.17T + 79T^{2} \) |
| 83 | \( 1 - 7.24iT - 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 5.31iT - 97T^{2} \) |
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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
−8.335260768958417290652301397506, −7.53060989856963751990739961811, −6.91778321127974191312723023602, −6.53693981865970585301711368782, −5.02305380204154814359238493532, −4.44100920473747083326182745437, −3.98857062500583237346671802458, −2.77540860423766843325473006515, −1.58983168386881278490549862253, −0.38802477178449291934993507152,
1.38836232336056791425888786262, 2.50036325789804702777280996256, 3.03856675937573385754030211960, 4.40955045838909837199829351213, 5.15463998899407528017326352293, 6.01368472841680434028342972522, 6.37321365231224452336056526052, 7.64175154454838633702975763648, 8.324160015281021285984618137939, 8.542917944620498670269800672090