| L(s) = 1 | − 4·5-s + 7-s − 3·9-s + 4·17-s + 6·25-s − 4·35-s − 12·37-s + 4·41-s + 12·45-s + 16·47-s + 49-s − 3·63-s − 16·67-s + 9·81-s + 16·83-s − 16·85-s − 28·89-s + 12·101-s + 12·109-s + 4·119-s + 6·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.377·7-s − 9-s + 0.970·17-s + 6/5·25-s − 0.676·35-s − 1.97·37-s + 0.624·41-s + 1.78·45-s + 2.33·47-s + 1/7·49-s − 0.377·63-s − 1.95·67-s + 81-s + 1.75·83-s − 1.73·85-s − 2.96·89-s + 1.19·101-s + 1.14·109-s + 0.366·119-s + 6/11·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.961079184601086706910241253566, −7.67548686778164902541947714763, −7.19238070289366458529053232147, −7.01210736968122329390212627163, −6.04594958296028195383672632634, −5.84527762283557267382239178151, −5.23638822530898309143358550273, −4.78746766862293010735127396082, −4.14883382526167974463210504827, −3.80052781215690022162466567950, −3.26692381730491515376569041297, −2.80939329027092828978550205754, −1.96324645037111540617948433860, −0.933679224635034246966499490286, 0,
0.933679224635034246966499490286, 1.96324645037111540617948433860, 2.80939329027092828978550205754, 3.26692381730491515376569041297, 3.80052781215690022162466567950, 4.14883382526167974463210504827, 4.78746766862293010735127396082, 5.23638822530898309143358550273, 5.84527762283557267382239178151, 6.04594958296028195383672632634, 7.01210736968122329390212627163, 7.19238070289366458529053232147, 7.67548686778164902541947714763, 7.961079184601086706910241253566
