| L(s) = 1 | + 3-s − 2·9-s − 3·17-s − 8·19-s + 25-s − 5·27-s − 6·41-s + 7·43-s − 10·49-s − 3·51-s − 8·57-s + 6·59-s − 5·67-s − 5·73-s + 75-s + 81-s − 6·83-s + 6·89-s − 14·97-s − 21·107-s + 15·113-s − 13·121-s − 6·123-s + 127-s + 7·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s − 0.727·17-s − 1.83·19-s + 1/5·25-s − 0.962·27-s − 0.937·41-s + 1.06·43-s − 1.42·49-s − 0.420·51-s − 1.05·57-s + 0.781·59-s − 0.610·67-s − 0.585·73-s + 0.115·75-s + 1/9·81-s − 0.658·83-s + 0.635·89-s − 1.42·97-s − 2.03·107-s + 1.41·113-s − 1.18·121-s − 0.541·123-s + 0.0887·127-s + 0.616·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 79 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−8.808087342157990609598333897972, −8.629425778415297704126623705589, −8.023417514441700227736394595987, −7.71487349820436593266884870103, −6.89401170141492015002059590364, −6.56363411735597423182665967724, −6.06726059439884025296949100825, −5.45708364438410569201652256800, −4.84546171896868340532130683728, −4.20460387184592916198747161147, −3.76249104604631907930161838454, −2.90325551029778057874916041022, −2.42732354279981849263841153213, −1.66025387856584383684798843751, 0,
1.66025387856584383684798843751, 2.42732354279981849263841153213, 2.90325551029778057874916041022, 3.76249104604631907930161838454, 4.20460387184592916198747161147, 4.84546171896868340532130683728, 5.45708364438410569201652256800, 6.06726059439884025296949100825, 6.56363411735597423182665967724, 6.89401170141492015002059590364, 7.71487349820436593266884870103, 8.023417514441700227736394595987, 8.629425778415297704126623705589, 8.808087342157990609598333897972
