| L(s) = 1 | − 3-s − 5-s − 2·7-s − 4·9-s − 6·11-s − 2·13-s + 15-s + 2·17-s − 8·19-s + 2·21-s − 2·23-s − 8·25-s + 6·27-s + 2·29-s − 3·31-s + 6·33-s + 2·35-s − 8·37-s + 2·39-s + 11·41-s − 13·43-s + 4·45-s − 8·47-s + 3·49-s − 2·51-s − 3·53-s + 6·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s − 4/3·9-s − 1.80·11-s − 0.554·13-s + 0.258·15-s + 0.485·17-s − 1.83·19-s + 0.436·21-s − 0.417·23-s − 8/5·25-s + 1.15·27-s + 0.371·29-s − 0.538·31-s + 1.04·33-s + 0.338·35-s − 1.31·37-s + 0.320·39-s + 1.71·41-s − 1.98·43-s + 0.596·45-s − 1.16·47-s + 3/7·49-s − 0.280·51-s − 0.412·53-s + 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 226576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 226576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 3 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 11 T + 81 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 13 T + 97 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T - 43 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 15 T + 159 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 21 T + 245 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 19 T + 253 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
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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
−10.62629017856199642801307602935, −10.51252311125244045988824095384, −9.966645345098210817996969322645, −9.638768119671122668957546664107, −8.728967377854870771040822927766, −8.658196690909550296204961145041, −7.87434135982152048716205109674, −7.82326062732188137232193230823, −7.15057066281716292898774468567, −6.37883853304271120249831911041, −6.10306310765673588210428179084, −5.69838754641572995579623865525, −4.93440284578987461429777565069, −4.83915503203818596764524072433, −3.72779820241498085148214794111, −3.37199028118464687204893382458, −2.54553834586009147071396942571, −2.10526407077591860907158640357, 0, 0,
2.10526407077591860907158640357, 2.54553834586009147071396942571, 3.37199028118464687204893382458, 3.72779820241498085148214794111, 4.83915503203818596764524072433, 4.93440284578987461429777565069, 5.69838754641572995579623865525, 6.10306310765673588210428179084, 6.37883853304271120249831911041, 7.15057066281716292898774468567, 7.82326062732188137232193230823, 7.87434135982152048716205109674, 8.658196690909550296204961145041, 8.728967377854870771040822927766, 9.638768119671122668957546664107, 9.966645345098210817996969322645, 10.51252311125244045988824095384, 10.62629017856199642801307602935
