L(s) = 1 | − 2-s + 4-s − 8-s − 3·9-s + 4·11-s − 3·13-s + 16-s − 17-s + 3·18-s − 4·22-s + 23-s + 3·26-s + 4·31-s − 32-s + 34-s − 3·36-s − 11·37-s + 10·41-s + 2·43-s + 4·44-s − 46-s + 11·47-s − 3·52-s − 53-s + 8·59-s + 8·61-s − 4·62-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 9-s + 1.20·11-s − 0.832·13-s + 1/4·16-s − 0.242·17-s + 0.707·18-s − 0.852·22-s + 0.208·23-s + 0.588·26-s + 0.718·31-s − 0.176·32-s + 0.171·34-s − 1/2·36-s − 1.80·37-s + 1.56·41-s + 0.304·43-s + 0.603·44-s − 0.147·46-s + 1.60·47-s − 0.416·52-s − 0.137·53-s + 1.04·59-s + 1.02·61-s − 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.66486563306522, −14.14613773049607, −13.87870496556819, −13.04094349652634, −12.38747465359559, −12.00347565812698, −11.58258052107753, −11.05551310619002, −10.53892659882311, −9.932712500925060, −9.417286791080220, −8.887450104432504, −8.589865696513544, −7.965956899014067, −7.182123879952363, −6.954589001779276, −6.234051316674914, −5.671071625998016, −5.162955997763484, −4.244863261664662, −3.812949143171800, −2.829116064593464, −2.528833032049554, −1.624939711839506, −0.8653034275834763, 0,
0.8653034275834763, 1.624939711839506, 2.528833032049554, 2.829116064593464, 3.812949143171800, 4.244863261664662, 5.162955997763484, 5.671071625998016, 6.234051316674914, 6.954589001779276, 7.182123879952363, 7.965956899014067, 8.589865696513544, 8.887450104432504, 9.417286791080220, 9.932712500925060, 10.53892659882311, 11.05551310619002, 11.58258052107753, 12.00347565812698, 12.38747465359559, 13.04094349652634, 13.87870496556819, 14.14613773049607, 14.66486563306522