L(s) = 1 | + 4-s − 9-s + 16-s − 8·23-s − 25-s − 6·29-s − 36-s − 8·37-s − 12·43-s − 8·53-s + 64-s + 20·67-s − 4·71-s − 20·79-s − 8·81-s − 8·92-s − 100-s + 12·107-s − 2·109-s + 8·113-s − 6·116-s − 18·121-s + 127-s + 131-s + 137-s + 139-s − 144-s + ⋯ |
L(s) = 1 | + 1/2·4-s − 1/3·9-s + 1/4·16-s − 1.66·23-s − 1/5·25-s − 1.11·29-s − 1/6·36-s − 1.31·37-s − 1.82·43-s − 1.09·53-s + 1/8·64-s + 2.44·67-s − 0.474·71-s − 2.25·79-s − 8/9·81-s − 0.834·92-s − 0.0999·100-s + 1.16·107-s − 0.191·109-s + 0.752·113-s − 0.557·116-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 117 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 103 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 129 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
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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
−8.678831705523340326545652473700, −8.224416239572792359446756199858, −7.84477582178301860280207781235, −7.33352489227896824982464612877, −6.74791701499940997518383557896, −6.44315517577070774610233060997, −5.71499191440762615317685767901, −5.49080639144099597772671774262, −4.79453071199600002822081908040, −4.07891098452839660483206604097, −3.55979711099246705792415994287, −2.98652964760512358792167314449, −2.09578724899994332343744223306, −1.62025429883584239177136663093, 0,
1.62025429883584239177136663093, 2.09578724899994332343744223306, 2.98652964760512358792167314449, 3.55979711099246705792415994287, 4.07891098452839660483206604097, 4.79453071199600002822081908040, 5.49080639144099597772671774262, 5.71499191440762615317685767901, 6.44315517577070774610233060997, 6.74791701499940997518383557896, 7.33352489227896824982464612877, 7.84477582178301860280207781235, 8.224416239572792359446756199858, 8.678831705523340326545652473700