| L(s) = 1 | − 2·5-s + 9-s − 2·13-s − 4·17-s − 3·25-s + 2·29-s + 8·37-s + 8·41-s − 2·45-s + 6·49-s − 6·53-s + 4·61-s + 4·65-s − 8·73-s − 8·81-s + 8·85-s − 24·89-s − 24·97-s − 16·101-s − 10·109-s + 12·113-s − 2·117-s + 121-s + 10·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1/3·9-s − 0.554·13-s − 0.970·17-s − 3/5·25-s + 0.371·29-s + 1.31·37-s + 1.24·41-s − 0.298·45-s + 6/7·49-s − 0.824·53-s + 0.512·61-s + 0.496·65-s − 0.936·73-s − 8/9·81-s + 0.867·85-s − 2.54·89-s − 2.43·97-s − 1.59·101-s − 0.957·109-s + 1.12·113-s − 0.184·117-s + 1/11·121-s + 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 215296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.710167446816163730450500525575, −8.285972470632452910770029277128, −7.890547307240037235736217368696, −7.29452121562612470815662658589, −7.10422671882119130255565536237, −6.41957940043265186380915690232, −5.89974333291319381524391102974, −5.34845078339962179578211488988, −4.58109700484804004381683902442, −4.21017420603606224423133595188, −3.86691781492097160093069142285, −2.84758136827322850409353999438, −2.43463857102373395387877195536, −1.32668373531693145636029385587, 0,
1.32668373531693145636029385587, 2.43463857102373395387877195536, 2.84758136827322850409353999438, 3.86691781492097160093069142285, 4.21017420603606224423133595188, 4.58109700484804004381683902442, 5.34845078339962179578211488988, 5.89974333291319381524391102974, 6.41957940043265186380915690232, 7.10422671882119130255565536237, 7.29452121562612470815662658589, 7.890547307240037235736217368696, 8.285972470632452910770029277128, 8.710167446816163730450500525575
