| L(s) = 1 | + 2·5-s − 2·7-s + 11-s + 2·13-s + 5·17-s − 2·19-s − 7·25-s + 29-s − 9·31-s − 4·35-s − 10·37-s + 3·41-s − 4·43-s − 12·47-s + 3·49-s + 11·53-s + 2·55-s + 2·59-s − 10·61-s + 4·65-s − 15·67-s − 18·71-s − 5·73-s − 2·77-s + 12·79-s − 13·83-s + 10·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s + 0.301·11-s + 0.554·13-s + 1.21·17-s − 0.458·19-s − 7/5·25-s + 0.185·29-s − 1.61·31-s − 0.676·35-s − 1.64·37-s + 0.468·41-s − 0.609·43-s − 1.75·47-s + 3/7·49-s + 1.51·53-s + 0.269·55-s + 0.260·59-s − 1.28·61-s + 0.496·65-s − 1.83·67-s − 2.13·71-s − 0.585·73-s − 0.227·77-s + 1.35·79-s − 1.42·83-s + 1.08·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + 39 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - T + 27 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9 T + 51 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 79 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 83 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 11 T + 105 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 39 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 15 T + 129 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 141 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 13 T + 207 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 198 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 18 T + 255 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.40152582777729722643150357514, −7.23037001837844469177723035797, −6.69354466685113406065587045649, −6.59017708368754238040335529643, −6.01077809559825820830113555828, −5.79682432142430521420368350166, −5.60309739469840712290416154037, −5.36917244988178119673272259977, −4.57341430749554621725338699665, −4.54904260799607834837252588195, −3.80926838283105878346505361352, −3.65269038143831425623329880508, −3.16613908814862365651645798412, −3.04575601033267448144181808018, −2.12131693164040254418867673376, −2.10263257003758193074506781278, −1.32227819317850434602707913526, −1.31519522073904260201629726352, 0, 0,
1.31519522073904260201629726352, 1.32227819317850434602707913526, 2.10263257003758193074506781278, 2.12131693164040254418867673376, 3.04575601033267448144181808018, 3.16613908814862365651645798412, 3.65269038143831425623329880508, 3.80926838283105878346505361352, 4.54904260799607834837252588195, 4.57341430749554621725338699665, 5.36917244988178119673272259977, 5.60309739469840712290416154037, 5.79682432142430521420368350166, 6.01077809559825820830113555828, 6.59017708368754238040335529643, 6.69354466685113406065587045649, 7.23037001837844469177723035797, 7.40152582777729722643150357514
