L(s) = 1 | − 3-s + 5-s − 7-s + 9-s + 2·13-s − 15-s − 6·17-s − 8·19-s + 21-s + 25-s − 27-s + 6·29-s + 4·31-s − 35-s − 10·37-s − 2·39-s − 6·41-s + 4·43-s + 45-s + 49-s + 6·51-s − 6·53-s + 8·57-s + 12·59-s − 10·61-s − 63-s + 2·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.258·15-s − 1.45·17-s − 1.83·19-s + 0.218·21-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.169·35-s − 1.64·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s + 1/7·49-s + 0.840·51-s − 0.824·53-s + 1.05·57-s + 1.56·59-s − 1.28·61-s − 0.125·63-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.746285736623812504733188648029, −8.508034972112431052168313360871, −7.02520028171832103268962093206, −6.50849900305861081354586811373, −5.88252979346856351246550317294, −4.76605760871970950359470598236, −4.08388045241890014835450076278, −2.75293092752386668281952313918, −1.65590098700280350442148076029, 0,
1.65590098700280350442148076029, 2.75293092752386668281952313918, 4.08388045241890014835450076278, 4.76605760871970950359470598236, 5.88252979346856351246550317294, 6.50849900305861081354586811373, 7.02520028171832103268962093206, 8.508034972112431052168313360871, 8.746285736623812504733188648029