| L(s) = 1 | − 2·3-s + 2·5-s + 4·7-s + 2·13-s − 4·15-s − 8·21-s + 3·25-s + 2·27-s − 4·31-s + 8·35-s − 10·37-s − 4·39-s + 12·41-s + 4·43-s + 12·47-s − 2·49-s + 18·53-s + 4·61-s + 4·65-s − 10·67-s + 12·71-s − 8·73-s − 6·75-s + 8·79-s − 81-s + 24·89-s + 8·91-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 1.51·7-s + 0.554·13-s − 1.03·15-s − 1.74·21-s + 3/5·25-s + 0.384·27-s − 0.718·31-s + 1.35·35-s − 1.64·37-s − 0.640·39-s + 1.87·41-s + 0.609·43-s + 1.75·47-s − 2/7·49-s + 2.47·53-s + 0.512·61-s + 0.496·65-s − 1.22·67-s + 1.42·71-s − 0.936·73-s − 0.692·75-s + 0.900·79-s − 1/9·81-s + 2.54·89-s + 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52128400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52128400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.113885583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.113885583\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 24 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 96 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 18 T + 184 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 156 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 154 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 24 T + 310 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T - 48 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−7.940162312285663301759172457539, −7.81532340263642971274507653877, −7.33157779994753733986000346525, −7.06847443341232265053085334201, −6.52054034537232450269114300559, −6.36770711573978749714756356731, −5.73367589085387466959861853298, −5.69885285052986013869079565906, −5.29233199099174526529636884709, −5.21260234148208233422336456837, −4.59107907412070758447498710536, −4.30436106492741519547448455270, −3.83203887611251216185402392045, −3.49425578025856736514961699931, −2.69359936318923714376757866550, −2.47702267706039017867841470517, −1.82894382954289172417351063329, −1.64021283516149235189977182489, −0.906319838568966220127028834704, −0.56286812921932456160399061625,
0.56286812921932456160399061625, 0.906319838568966220127028834704, 1.64021283516149235189977182489, 1.82894382954289172417351063329, 2.47702267706039017867841470517, 2.69359936318923714376757866550, 3.49425578025856736514961699931, 3.83203887611251216185402392045, 4.30436106492741519547448455270, 4.59107907412070758447498710536, 5.21260234148208233422336456837, 5.29233199099174526529636884709, 5.69885285052986013869079565906, 5.73367589085387466959861853298, 6.36770711573978749714756356731, 6.52054034537232450269114300559, 7.06847443341232265053085334201, 7.33157779994753733986000346525, 7.81532340263642971274507653877, 7.940162312285663301759172457539
