| L(s) = 1 | + 2·3-s + 3·9-s + 8·11-s − 4·17-s + 8·19-s − 2·25-s + 4·27-s + 16·33-s − 20·41-s + 24·43-s − 6·49-s − 8·51-s + 16·57-s − 8·59-s + 8·67-s + 4·73-s − 4·75-s + 5·81-s − 8·83-s − 12·89-s + 28·97-s + 24·99-s + 24·107-s + 4·113-s + 26·121-s − 40·123-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 2.41·11-s − 0.970·17-s + 1.83·19-s − 2/5·25-s + 0.769·27-s + 2.78·33-s − 3.12·41-s + 3.65·43-s − 6/7·49-s − 1.12·51-s + 2.11·57-s − 1.04·59-s + 0.977·67-s + 0.468·73-s − 0.461·75-s + 5/9·81-s − 0.878·83-s − 1.27·89-s + 2.84·97-s + 2.41·99-s + 2.32·107-s + 0.376·113-s + 2.36·121-s − 3.60·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.846792627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.846792627\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−10.28009841344593893076259941434, −10.01654175763543856388739183935, −9.503604753036850784671055094742, −9.212078102163279702311731705058, −8.812090365385096052225426965659, −8.731262824133501400138219398389, −7.81343294487424868030858234002, −7.69998266941098431523815804069, −6.97543633674586192712503611084, −6.82565623673539050240333032784, −6.22865905640795630079773976967, −5.79682295214575974826715623297, −4.97470018237362601528702752031, −4.56637920958706365069482256691, −3.88639137135655259937820080149, −3.64457877857498921465958647906, −3.11104644282753822814754116264, −2.33996003394886694012888149438, −1.64087392702100692375591053739, −1.06157146329068855737319521970,
1.06157146329068855737319521970, 1.64087392702100692375591053739, 2.33996003394886694012888149438, 3.11104644282753822814754116264, 3.64457877857498921465958647906, 3.88639137135655259937820080149, 4.56637920958706365069482256691, 4.97470018237362601528702752031, 5.79682295214575974826715623297, 6.22865905640795630079773976967, 6.82565623673539050240333032784, 6.97543633674586192712503611084, 7.69998266941098431523815804069, 7.81343294487424868030858234002, 8.731262824133501400138219398389, 8.812090365385096052225426965659, 9.212078102163279702311731705058, 9.503604753036850784671055094742, 10.01654175763543856388739183935, 10.28009841344593893076259941434
