| L(s) = 1 | − 2-s − 4-s − 7-s + 3·8-s + 11-s + 2·13-s + 14-s − 16-s + 2·17-s + 4·19-s − 22-s − 2·26-s + 28-s − 6·29-s − 5·32-s − 2·34-s − 6·37-s − 4·38-s + 6·41-s + 4·43-s − 44-s + 49-s − 2·52-s − 2·53-s − 3·56-s + 6·58-s − 4·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.377·7-s + 1.06·8-s + 0.301·11-s + 0.554·13-s + 0.267·14-s − 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.213·22-s − 0.392·26-s + 0.188·28-s − 1.11·29-s − 0.883·32-s − 0.342·34-s − 0.986·37-s − 0.648·38-s + 0.937·41-s + 0.609·43-s − 0.150·44-s + 1/7·49-s − 0.277·52-s − 0.274·53-s − 0.400·56-s + 0.787·58-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−16.16561092274675, −15.84376115422444, −15.02446509864542, −14.40488170902534, −13.91169681303963, −13.41862139427361, −12.86488043260328, −12.27326640737938, −11.61933462764519, −10.92996795000723, −10.48668339494477, −9.717001270823101, −9.432643842934386, −8.817127353974364, −8.297820378817258, −7.458494772209010, −7.273069816483390, −6.250247132794422, −5.639171271422863, −5.023179394339256, −4.138226591057971, −3.643645877353267, −2.836673361334795, −1.682690395376015, −1.035173734102012, 0,
1.035173734102012, 1.682690395376015, 2.836673361334795, 3.643645877353267, 4.138226591057971, 5.023179394339256, 5.639171271422863, 6.250247132794422, 7.273069816483390, 7.458494772209010, 8.297820378817258, 8.817127353974364, 9.432643842934386, 9.717001270823101, 10.48668339494477, 10.92996795000723, 11.61933462764519, 12.27326640737938, 12.86488043260328, 13.41862139427361, 13.91169681303963, 14.40488170902534, 15.02446509864542, 15.84376115422444, 16.16561092274675
