| L(s) = 1 | + 5-s + 9-s − 8·17-s + 25-s − 6·29-s − 8·37-s − 4·41-s + 45-s − 2·49-s − 12·61-s − 6·73-s + 81-s − 8·85-s + 8·89-s − 18·97-s + 2·101-s − 8·109-s − 4·113-s − 6·121-s + 125-s + 127-s + 131-s + 137-s + 139-s − 6·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1/3·9-s − 1.94·17-s + 1/5·25-s − 1.11·29-s − 1.31·37-s − 0.624·41-s + 0.149·45-s − 2/7·49-s − 1.53·61-s − 0.702·73-s + 1/9·81-s − 0.867·85-s + 0.847·89-s − 1.82·97-s + 0.199·101-s − 0.766·109-s − 0.376·113-s − 0.545·121-s + 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.498·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( 1 - T \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−8.715401824288490484091411561854, −8.268102337831885191220892813782, −7.61070364154779204847846797552, −7.16220222268533246083565567216, −6.70985641802845926680135362376, −6.34698728864653666113246245556, −5.73396733441215755020262261236, −5.20936382338881249065268750246, −4.66402724102244385136976031673, −4.18302302984365780653215522776, −3.55106093384866926465866439540, −2.82884458062098681050047804092, −2.07206124622038359909226857680, −1.55191890381928017668501542423, 0,
1.55191890381928017668501542423, 2.07206124622038359909226857680, 2.82884458062098681050047804092, 3.55106093384866926465866439540, 4.18302302984365780653215522776, 4.66402724102244385136976031673, 5.20936382338881249065268750246, 5.73396733441215755020262261236, 6.34698728864653666113246245556, 6.70985641802845926680135362376, 7.16220222268533246083565567216, 7.61070364154779204847846797552, 8.268102337831885191220892813782, 8.715401824288490484091411561854
