| L(s) = 1 | − 2·3-s + 2·5-s + 2·7-s − 9-s − 4·13-s − 4·15-s + 4·19-s − 4·21-s + 4·23-s + 3·25-s + 6·27-s − 4·29-s + 4·35-s − 8·37-s + 8·39-s + 2·41-s + 10·43-s − 2·45-s − 18·47-s − 9·49-s − 4·53-s − 8·57-s − 2·61-s − 2·63-s − 8·65-s + 10·67-s − 8·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 0.755·7-s − 1/3·9-s − 1.10·13-s − 1.03·15-s + 0.917·19-s − 0.872·21-s + 0.834·23-s + 3/5·25-s + 1.15·27-s − 0.742·29-s + 0.676·35-s − 1.31·37-s + 1.28·39-s + 0.312·41-s + 1.52·43-s − 0.298·45-s − 2.62·47-s − 9/7·49-s − 0.549·53-s − 1.05·57-s − 0.256·61-s − 0.251·63-s − 0.992·65-s + 1.22·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93702400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93702400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 11 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 109 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 18 T + 173 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 157 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 155 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.34592229563283193491100102117, −7.10050352974757718623387664199, −6.70001284495691808001784764380, −6.64108838898045297503782033631, −5.88250630001594965129510568388, −5.80558353300617983734879709967, −5.40064632878912385465737015058, −5.30334241520526876407659526589, −4.77887281040691582495575893442, −4.66500485957538367594639257405, −4.23354305160660310076416562911, −3.47076776298527134731772270622, −3.15897662787053467601991214869, −2.89146151581615367550715677658, −2.23029959074249509703685269383, −1.99058359999203636694300086501, −1.29565516038587435540964779964, −1.11699565031101077599152916895, 0, 0,
1.11699565031101077599152916895, 1.29565516038587435540964779964, 1.99058359999203636694300086501, 2.23029959074249509703685269383, 2.89146151581615367550715677658, 3.15897662787053467601991214869, 3.47076776298527134731772270622, 4.23354305160660310076416562911, 4.66500485957538367594639257405, 4.77887281040691582495575893442, 5.30334241520526876407659526589, 5.40064632878912385465737015058, 5.80558353300617983734879709967, 5.88250630001594965129510568388, 6.64108838898045297503782033631, 6.70001284495691808001784764380, 7.10050352974757718623387664199, 7.34592229563283193491100102117
