| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 4·11-s − 2·13-s + 16-s + 6·17-s + 4·19-s − 20-s + 4·22-s + 23-s + 25-s + 2·26-s + 2·29-s − 32-s − 6·34-s − 2·37-s − 4·38-s + 40-s − 10·41-s − 4·43-s − 4·44-s − 46-s − 7·49-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 1.20·11-s − 0.554·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s − 0.223·20-s + 0.852·22-s + 0.208·23-s + 1/5·25-s + 0.392·26-s + 0.371·29-s − 0.176·32-s − 1.02·34-s − 0.328·37-s − 0.648·38-s + 0.158·40-s − 1.56·41-s − 0.609·43-s − 0.603·44-s − 0.147·46-s − 49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.609088489832511873090115492570, −7.87701383658038213687540209690, −7.49581125142151687619554286458, −6.57997984311815878143326193704, −5.44841166558780959017176507263, −4.90036982975824370397498764774, −3.44221025217975119382003531721, −2.79062662384276445404170734102, −1.41543415021040645255010595248, 0,
1.41543415021040645255010595248, 2.79062662384276445404170734102, 3.44221025217975119382003531721, 4.90036982975824370397498764774, 5.44841166558780959017176507263, 6.57997984311815878143326193704, 7.49581125142151687619554286458, 7.87701383658038213687540209690, 8.609088489832511873090115492570
