| L(s) = 1 | + 3·9-s + 4·11-s − 12·29-s − 4·31-s + 20·41-s + 13·49-s − 16·59-s − 14·61-s + 32·71-s + 8·79-s + 9·81-s + 26·89-s + 12·99-s + 6·101-s + 18·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 20·169-s + 173-s + ⋯ |
| L(s) = 1 | + 9-s + 1.20·11-s − 2.22·29-s − 0.718·31-s + 3.12·41-s + 13/7·49-s − 2.08·59-s − 1.79·61-s + 3.79·71-s + 0.900·79-s + 81-s + 2.75·89-s + 1.20·99-s + 0.597·101-s + 1.72·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.53·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.855404530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.855404530\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 35 T^{2} + 696 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 10 T^{2} - 1269 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 30 T^{2} - 1309 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 59 | $C_2^2$ | \( ( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 125 T^{2} + 11136 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 50 T^{2} - 2829 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 165 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 13 T + 80 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.63793255811454605600635851568, −7.31817762504316894269345615549, −7.17221391813796075269955755591, −6.71159370016484315171196704011, −6.70824588255331453991686156353, −6.18295792692781301525212610270, −6.17608890009838146258827061146, −5.88896691527138101467521251661, −5.84711342882042746286338130168, −5.18872742206798954139342115549, −5.18314219591605660247033744968, −4.87661561592478920467818965468, −4.59818711338232673591791562115, −4.15013238338767560023104257017, −4.06916457450381108752699357260, −3.78097153947895385409843541218, −3.72783207606035734724301947734, −3.11553387377460623112050030319, −3.10169379256012402887933854753, −2.37389021079181415499955452139, −2.21874116559189075748288815535, −1.72196293071766099245710907694, −1.70756452577681252597114183710, −0.825344116128527881505054229628, −0.71750846692866205821574959380,
0.71750846692866205821574959380, 0.825344116128527881505054229628, 1.70756452577681252597114183710, 1.72196293071766099245710907694, 2.21874116559189075748288815535, 2.37389021079181415499955452139, 3.10169379256012402887933854753, 3.11553387377460623112050030319, 3.72783207606035734724301947734, 3.78097153947895385409843541218, 4.06916457450381108752699357260, 4.15013238338767560023104257017, 4.59818711338232673591791562115, 4.87661561592478920467818965468, 5.18314219591605660247033744968, 5.18872742206798954139342115549, 5.84711342882042746286338130168, 5.88896691527138101467521251661, 6.17608890009838146258827061146, 6.18295792692781301525212610270, 6.70824588255331453991686156353, 6.71159370016484315171196704011, 7.17221391813796075269955755591, 7.31817762504316894269345615549, 7.63793255811454605600635851568
