| L(s) = 1 | + 2·5-s − 4·7-s + 4·13-s + 2·19-s + 6·23-s + 3·25-s − 4·31-s − 8·35-s + 16·37-s − 4·43-s + 6·47-s + 49-s − 18·53-s + 4·61-s + 8·65-s − 16·67-s + 12·71-s − 8·73-s + 8·79-s + 24·83-s + 24·89-s − 16·91-s + 4·95-s + 16·97-s − 12·101-s + 20·103-s + 24·107-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s + 1.10·13-s + 0.458·19-s + 1.25·23-s + 3/5·25-s − 0.718·31-s − 1.35·35-s + 2.63·37-s − 0.609·43-s + 0.875·47-s + 1/7·49-s − 2.47·53-s + 0.512·61-s + 0.992·65-s − 1.95·67-s + 1.42·71-s − 0.936·73-s + 0.900·79-s + 2.63·83-s + 2.54·89-s − 1.67·91-s + 0.410·95-s + 1.62·97-s − 1.19·101-s + 1.97·103-s + 2.32·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41990400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41990400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.861118771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.861118771\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T - 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 91 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 18 T + 175 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 24 T + 298 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 210 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−8.197285997615300830375974048381, −7.81954393643078066454034022857, −7.34931342603331488401936437746, −7.22699076806315844693263866293, −6.51468662645145753252418346731, −6.41697480664872051245282984620, −6.07082437892846404404910559051, −5.99243939984286688903532971375, −5.43431451567671090978770380573, −4.84663124938125165389894799694, −4.78929914382997569730204983240, −4.21796911302626611004598862950, −3.49988654084750094469624131866, −3.43358109148721695463216454045, −3.05528812677609273491182678646, −2.65193786923731171444655459082, −1.99381315136792885981583950205, −1.68679312723488129557104913657, −0.813472835380244922495323187130, −0.66188319967947064809871692706,
0.66188319967947064809871692706, 0.813472835380244922495323187130, 1.68679312723488129557104913657, 1.99381315136792885981583950205, 2.65193786923731171444655459082, 3.05528812677609273491182678646, 3.43358109148721695463216454045, 3.49988654084750094469624131866, 4.21796911302626611004598862950, 4.78929914382997569730204983240, 4.84663124938125165389894799694, 5.43431451567671090978770380573, 5.99243939984286688903532971375, 6.07082437892846404404910559051, 6.41697480664872051245282984620, 6.51468662645145753252418346731, 7.22699076806315844693263866293, 7.34931342603331488401936437746, 7.81954393643078066454034022857, 8.197285997615300830375974048381
