| L(s) = 1 | − 4-s − 2·5-s − 2·7-s + 4·13-s − 3·16-s − 2·19-s + 2·20-s − 6·23-s + 3·25-s + 2·28-s − 2·31-s + 4·35-s − 8·37-s + 22·43-s − 12·47-s − 8·49-s − 4·52-s − 18·59-s + 4·61-s + 7·64-s − 8·65-s − 8·67-s − 12·71-s + 4·73-s + 2·76-s + 4·79-s + 6·80-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.894·5-s − 0.755·7-s + 1.10·13-s − 3/4·16-s − 0.458·19-s + 0.447·20-s − 1.25·23-s + 3/5·25-s + 0.377·28-s − 0.359·31-s + 0.676·35-s − 1.31·37-s + 3.35·43-s − 1.75·47-s − 8/7·49-s − 0.554·52-s − 2.34·59-s + 0.512·61-s + 7/8·64-s − 0.992·65-s − 0.977·67-s − 1.42·71-s + 0.468·73-s + 0.229·76-s + 0.450·79-s + 0.670·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16040025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16040025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 89 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 22 T + 204 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 18 T + 172 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
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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
−8.159152338564646011676389353880, −8.065913773794649636757084844804, −7.52051863314098434274761163948, −7.23451318283447488787265284261, −6.84500267428899929937837977230, −6.24200796805858490281556714860, −6.12879894922061745591756806968, −5.91087609359895699423617623650, −5.12439884038805717627618144021, −4.83436781212797976038073497539, −4.35275786591293854552655265013, −4.08714628945855481481515408270, −3.59451150419105758063568707968, −3.39423268664610769756892939624, −2.82287087072538886729844440393, −2.26138112958945799395528573654, −1.66898086325891710742893623965, −1.04961533659900606996507482716, 0, 0,
1.04961533659900606996507482716, 1.66898086325891710742893623965, 2.26138112958945799395528573654, 2.82287087072538886729844440393, 3.39423268664610769756892939624, 3.59451150419105758063568707968, 4.08714628945855481481515408270, 4.35275786591293854552655265013, 4.83436781212797976038073497539, 5.12439884038805717627618144021, 5.91087609359895699423617623650, 6.12879894922061745591756806968, 6.24200796805858490281556714860, 6.84500267428899929937837977230, 7.23451318283447488787265284261, 7.52051863314098434274761163948, 8.065913773794649636757084844804, 8.159152338564646011676389353880
