| L(s) = 1 | − 2·3-s + 3·9-s − 4·11-s − 8·23-s − 10·25-s − 4·27-s − 4·31-s + 8·33-s + 12·37-s − 8·47-s + 6·49-s + 8·53-s + 8·59-s + 8·67-s + 16·69-s + 8·71-s + 20·75-s + 5·81-s − 4·89-s + 8·93-s + 12·97-s − 12·99-s − 12·103-s − 24·111-s − 4·113-s + 5·121-s + 127-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s − 1.20·11-s − 1.66·23-s − 2·25-s − 0.769·27-s − 0.718·31-s + 1.39·33-s + 1.97·37-s − 1.16·47-s + 6/7·49-s + 1.09·53-s + 1.04·59-s + 0.977·67-s + 1.92·69-s + 0.949·71-s + 2.30·75-s + 5/9·81-s − 0.423·89-s + 0.829·93-s + 1.21·97-s − 1.20·99-s − 1.18·103-s − 2.27·111-s − 0.376·113-s + 5/11·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.07641104884056435139819806889, −6.73638794275886190021986641926, −6.19951579414602576430835795528, −5.88213205104872133241524618721, −5.57173953373637205295252912936, −5.25918397348436745237982895745, −4.74562682915512166148308984470, −4.12188479494691834847210109401, −3.96990974048973940554899541664, −3.42107966854588931282020215251, −2.46913686410446462838923871373, −2.27948613892252072252465087023, −1.60965270260187357525323612604, −0.67942463774858439823406722310, 0,
0.67942463774858439823406722310, 1.60965270260187357525323612604, 2.27948613892252072252465087023, 2.46913686410446462838923871373, 3.42107966854588931282020215251, 3.96990974048973940554899541664, 4.12188479494691834847210109401, 4.74562682915512166148308984470, 5.25918397348436745237982895745, 5.57173953373637205295252912936, 5.88213205104872133241524618721, 6.19951579414602576430835795528, 6.73638794275886190021986641926, 7.07641104884056435139819806889
