L(s) = 1 | + 2·3-s + 3·9-s − 8·11-s − 12·17-s − 8·19-s + 25-s + 4·27-s − 16·33-s − 12·41-s + 24·43-s − 14·49-s − 24·51-s − 16·57-s + 24·59-s + 8·67-s − 12·73-s + 2·75-s + 5·81-s − 24·83-s + 20·89-s + 4·97-s − 24·99-s − 8·107-s − 12·113-s + 26·121-s − 24·123-s + 127-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 9-s − 2.41·11-s − 2.91·17-s − 1.83·19-s + 1/5·25-s + 0.769·27-s − 2.78·33-s − 1.87·41-s + 3.65·43-s − 2·49-s − 3.36·51-s − 2.11·57-s + 3.12·59-s + 0.977·67-s − 1.40·73-s + 0.230·75-s + 5/9·81-s − 2.63·83-s + 2.11·89-s + 0.406·97-s − 2.41·99-s − 0.773·107-s − 1.12·113-s + 2.36·121-s − 2.16·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.042960162718976648409639393560, −8.645810177900386367635479957201, −8.323057177226078640880314002796, −7.982386705736041440880055703756, −7.23083008412003813869292936859, −6.89270810769313104844169224865, −6.33640038439988253692354509575, −5.59585158632682227968586342616, −4.88793662384328778723790668153, −4.41954513526927827595737015950, −3.95303834850677407956821293662, −2.92311775669320789657547583777, −2.25002432004047770108316173263, −2.22526038506206736624914155682, 0,
2.22526038506206736624914155682, 2.25002432004047770108316173263, 2.92311775669320789657547583777, 3.95303834850677407956821293662, 4.41954513526927827595737015950, 4.88793662384328778723790668153, 5.59585158632682227968586342616, 6.33640038439988253692354509575, 6.89270810769313104844169224865, 7.23083008412003813869292936859, 7.982386705736041440880055703756, 8.323057177226078640880314002796, 8.645810177900386367635479957201, 9.042960162718976648409639393560