| L(s) = 1 | − 5-s − 2·7-s + 4·11-s + 3·13-s − 2·19-s − 2·23-s − 8·25-s − 2·29-s − 8·31-s + 2·35-s − 2·37-s − 6·41-s − 5·43-s + 6·47-s + 3·49-s + 3·53-s − 4·55-s + 9·59-s − 5·61-s − 3·65-s − 9·67-s + 15·71-s − 14·73-s − 8·77-s − 12·79-s − 2·83-s − 89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 1.20·11-s + 0.832·13-s − 0.458·19-s − 0.417·23-s − 8/5·25-s − 0.371·29-s − 1.43·31-s + 0.338·35-s − 0.328·37-s − 0.937·41-s − 0.762·43-s + 0.875·47-s + 3/7·49-s + 0.412·53-s − 0.539·55-s + 1.17·59-s − 0.640·61-s − 0.372·65-s − 1.09·67-s + 1.78·71-s − 1.63·73-s − 0.911·77-s − 1.35·79-s − 0.219·83-s − 0.105·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33593616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33593616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 27 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 73 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5 T - 9 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 77 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 9 T + 137 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 67 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 9 T + 153 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 15 T + 187 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 14 T + 115 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 189 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 87 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + T + 167 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 193 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.73119143197182237632932747957, −7.68023797292528946968668234871, −7.08769180888866020306832254766, −7.04247334731434584267834566592, −6.35976922755658022643673471212, −6.22136110767779554527964356301, −5.92881786021943058699173344836, −5.48374327873897644378163079483, −5.11204390262409081135852632293, −4.52224220692400464280820159951, −4.09115416782106838140010803745, −3.82264561672961824029967557993, −3.44525355223830436917748517548, −3.40366368247158496781973535993, −2.33742968589576369343234633398, −2.28814013490426847377914888981, −1.39859860943924621972867820796, −1.27252270521648237557386659065, 0, 0,
1.27252270521648237557386659065, 1.39859860943924621972867820796, 2.28814013490426847377914888981, 2.33742968589576369343234633398, 3.40366368247158496781973535993, 3.44525355223830436917748517548, 3.82264561672961824029967557993, 4.09115416782106838140010803745, 4.52224220692400464280820159951, 5.11204390262409081135852632293, 5.48374327873897644378163079483, 5.92881786021943058699173344836, 6.22136110767779554527964356301, 6.35976922755658022643673471212, 7.04247334731434584267834566592, 7.08769180888866020306832254766, 7.68023797292528946968668234871, 7.73119143197182237632932747957
