| L(s) = 1 | − 2-s − 6·5-s + 8-s + 6·10-s + 12·11-s + 2·13-s − 16-s − 6·17-s − 7·19-s − 12·22-s − 6·23-s + 17·25-s − 2·26-s + 6·29-s + 2·31-s + 6·34-s − 2·37-s + 7·38-s − 6·40-s − 2·43-s + 6·46-s − 17·50-s + 6·53-s − 72·55-s − 6·58-s + 5·61-s − 2·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2.68·5-s + 0.353·8-s + 1.89·10-s + 3.61·11-s + 0.554·13-s − 1/4·16-s − 1.45·17-s − 1.60·19-s − 2.55·22-s − 1.25·23-s + 17/5·25-s − 0.392·26-s + 1.11·29-s + 0.359·31-s + 1.02·34-s − 0.328·37-s + 1.13·38-s − 0.948·40-s − 0.304·43-s + 0.884·46-s − 2.40·50-s + 0.824·53-s − 9.70·55-s − 0.787·58-s + 0.640·61-s − 0.254·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7243196340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7243196340\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.875789891962849293403792378435, −8.622826064219430318628867187726, −8.343897555668474574063861872666, −8.214070908011174031726282009713, −7.54371351064716783115683329774, −7.14040562688200470193188436231, −6.77739746049386281226645157403, −6.69464132667880455127242339186, −6.02889703033778877596416487259, −5.97183194512435554111946878815, −4.66501747363712042326041385704, −4.41876550517064804341344394664, −4.17149915107416829578982286192, −4.12028699536755786353381521402, −3.38820515104997372048400013766, −3.35198720576163006533059845225, −2.12761041558773500646144942325, −1.72859897867610761726232932864, −0.941059663090886205703040339871, −0.41718385293293338696864043095,
0.41718385293293338696864043095, 0.941059663090886205703040339871, 1.72859897867610761726232932864, 2.12761041558773500646144942325, 3.35198720576163006533059845225, 3.38820515104997372048400013766, 4.12028699536755786353381521402, 4.17149915107416829578982286192, 4.41876550517064804341344394664, 4.66501747363712042326041385704, 5.97183194512435554111946878815, 6.02889703033778877596416487259, 6.69464132667880455127242339186, 6.77739746049386281226645157403, 7.14040562688200470193188436231, 7.54371351064716783115683329774, 8.214070908011174031726282009713, 8.343897555668474574063861872666, 8.622826064219430318628867187726, 8.875789891962849293403792378435
