| L(s) = 1 | + 2·7-s + 10·13-s + 4·19-s − 7·25-s + 4·31-s − 14·37-s + 16·43-s + 3·49-s − 2·61-s + 4·67-s + 22·73-s + 28·79-s + 20·91-s + 4·97-s − 8·103-s + 22·109-s − 10·121-s + 127-s + 131-s + 8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 2.77·13-s + 0.917·19-s − 7/5·25-s + 0.718·31-s − 2.30·37-s + 2.43·43-s + 3/7·49-s − 0.256·61-s + 0.488·67-s + 2.57·73-s + 3.15·79-s + 2.09·91-s + 0.406·97-s − 0.788·103-s + 2.10·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.693·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.801720434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.801720434\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 175 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.072816093438087714260956216902, −8.816874804479432557322232197721, −8.338441837433368339445617442390, −8.238281701572889519417981810853, −7.59942029419832757322927791745, −7.53231763980010663747111648373, −6.81137645569507189023563203258, −6.44270235831495573613343682420, −6.03611429095438001809706493767, −5.78980953826140060719453181803, −5.15057417688175895617129518941, −5.06505742123967975275650776555, −4.07985423647331204119384268824, −4.07977990615109222406238488382, −3.35448989415524127485545985783, −3.32567588769675931081988500314, −2.20187330442277585990298382092, −1.96091800486394431434211509337, −1.15835922364910855749890096371, −0.799407479757812069684742056716,
0.799407479757812069684742056716, 1.15835922364910855749890096371, 1.96091800486394431434211509337, 2.20187330442277585990298382092, 3.32567588769675931081988500314, 3.35448989415524127485545985783, 4.07977990615109222406238488382, 4.07985423647331204119384268824, 5.06505742123967975275650776555, 5.15057417688175895617129518941, 5.78980953826140060719453181803, 6.03611429095438001809706493767, 6.44270235831495573613343682420, 6.81137645569507189023563203258, 7.53231763980010663747111648373, 7.59942029419832757322927791745, 8.238281701572889519417981810853, 8.338441837433368339445617442390, 8.816874804479432557322232197721, 9.072816093438087714260956216902
