| L(s) = 1 | + 4-s − 4·5-s − 3·9-s + 2·11-s + 16-s − 4·20-s + 10·23-s + 2·25-s − 3·36-s + 2·44-s + 12·45-s − 16·47-s − 5·49-s − 8·55-s + 64-s − 4·80-s + 10·92-s − 6·99-s + 2·100-s − 40·115-s − 7·121-s + 28·125-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.78·5-s − 9-s + 0.603·11-s + 1/4·16-s − 0.894·20-s + 2.08·23-s + 2/5·25-s − 1/2·36-s + 0.301·44-s + 1.78·45-s − 2.33·47-s − 5/7·49-s − 1.07·55-s + 1/8·64-s − 0.447·80-s + 1.04·92-s − 0.603·99-s + 1/5·100-s − 3.73·115-s − 0.636·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/4·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 19 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 91 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−8.790588601535390277292601816676, −8.428795589558927348932545064837, −7.932250930921357396197811411568, −7.59637740604431091607700513464, −7.04729226132818219446518683825, −6.59351405053505432939215471518, −6.11693789079955114728164002848, −5.35115169010777942506357811198, −4.85738801044108780443732649673, −4.25517082503526883883452937780, −3.45956340591956291735517152818, −3.32009790808247759240534168188, −2.49392875440254451170783957070, −1.31654176033133009955997278519, 0,
1.31654176033133009955997278519, 2.49392875440254451170783957070, 3.32009790808247759240534168188, 3.45956340591956291735517152818, 4.25517082503526883883452937780, 4.85738801044108780443732649673, 5.35115169010777942506357811198, 6.11693789079955114728164002848, 6.59351405053505432939215471518, 7.04729226132818219446518683825, 7.59637740604431091607700513464, 7.932250930921357396197811411568, 8.428795589558927348932545064837, 8.790588601535390277292601816676
