L(s) = 1 | + 9-s − 4·13-s − 4·16-s − 4·17-s − 6·23-s − 3·25-s − 10·29-s − 6·43-s − 49-s + 6·53-s + 22·79-s + 81-s − 8·101-s − 16·107-s + 26·113-s − 4·117-s + 6·121-s + 127-s + 131-s + 137-s + 139-s − 4·144-s + 149-s + 151-s − 4·153-s + 157-s + 163-s + ⋯ |
L(s) = 1 | + 1/3·9-s − 1.10·13-s − 16-s − 0.970·17-s − 1.25·23-s − 3/5·25-s − 1.85·29-s − 0.914·43-s − 1/7·49-s + 0.824·53-s + 2.47·79-s + 1/9·81-s − 0.796·101-s − 1.54·107-s + 2.44·113-s − 0.369·117-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/3·144-s + 0.0819·149-s + 0.0813·151-s − 0.323·153-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 87 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 153 T^{2} + p^{2} T^{4} \) |
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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
−9.658857411027721060650132921722, −9.104325020682672344238016194049, −8.617402224332648445287495409944, −7.87947844894776359702386421722, −7.60286082752122214635117467367, −6.88788192379235271768016916216, −6.61398730951695608876716735048, −5.83113586638032303860407128672, −5.28599986512932789916623660115, −4.60493279061637157579336474667, −4.12337519690384953946299270603, −3.44889424356380479348839351191, −2.29885484837432111545139589440, −1.96574544665697132650989518108, 0,
1.96574544665697132650989518108, 2.29885484837432111545139589440, 3.44889424356380479348839351191, 4.12337519690384953946299270603, 4.60493279061637157579336474667, 5.28599986512932789916623660115, 5.83113586638032303860407128672, 6.61398730951695608876716735048, 6.88788192379235271768016916216, 7.60286082752122214635117467367, 7.87947844894776359702386421722, 8.617402224332648445287495409944, 9.104325020682672344238016194049, 9.658857411027721060650132921722