| L(s) = 1 | + 4-s − 4·13-s + 16-s − 4·19-s − 25-s + 14·31-s − 20·37-s − 8·43-s − 4·52-s + 8·61-s + 64-s + 4·67-s − 4·73-s − 4·76-s + 10·79-s + 26·97-s − 100-s + 32·103-s − 20·109-s − 13·121-s + 14·124-s + 127-s + 131-s + 137-s + 139-s − 20·148-s + 149-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.10·13-s + 1/4·16-s − 0.917·19-s − 1/5·25-s + 2.51·31-s − 3.28·37-s − 1.21·43-s − 0.554·52-s + 1.02·61-s + 1/8·64-s + 0.488·67-s − 0.468·73-s − 0.458·76-s + 1.12·79-s + 2.63·97-s − 0.0999·100-s + 3.15·103-s − 1.91·109-s − 1.18·121-s + 1.25·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.64·148-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
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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
−8.092040072792470726203226030224, −7.56006473562539600008273040402, −7.04887098814165978645228240103, −6.76340153457321918972152567863, −6.33999540350047929222894736546, −5.88431208298112457486858305913, −5.10036072121782338740270472262, −4.93177667775637839170766051994, −4.42589737754936873558841343107, −3.52994453436524682105467711302, −3.35517260251051468988940553435, −2.28166780967037125049719495570, −2.26518482860346062093265617714, −1.22415852747718716380535497945, 0,
1.22415852747718716380535497945, 2.26518482860346062093265617714, 2.28166780967037125049719495570, 3.35517260251051468988940553435, 3.52994453436524682105467711302, 4.42589737754936873558841343107, 4.93177667775637839170766051994, 5.10036072121782338740270472262, 5.88431208298112457486858305913, 6.33999540350047929222894736546, 6.76340153457321918972152567863, 7.04887098814165978645228240103, 7.56006473562539600008273040402, 8.092040072792470726203226030224
