| L(s) = 1 | + 2·7-s + 4·11-s + 4·13-s − 8·17-s + 4·23-s + 2·25-s − 4·29-s + 8·31-s − 4·37-s − 16·41-s + 16·43-s + 8·47-s + 3·49-s + 4·53-s − 16·59-s − 4·61-s − 8·67-s + 12·71-s + 12·73-s + 8·77-s + 8·79-s − 8·83-s + 8·91-s − 4·97-s + 24·101-s + 8·103-s + 4·107-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 1.20·11-s + 1.10·13-s − 1.94·17-s + 0.834·23-s + 2/5·25-s − 0.742·29-s + 1.43·31-s − 0.657·37-s − 2.49·41-s + 2.43·43-s + 1.16·47-s + 3/7·49-s + 0.549·53-s − 2.08·59-s − 0.512·61-s − 0.977·67-s + 1.42·71-s + 1.40·73-s + 0.911·77-s + 0.900·79-s − 0.878·83-s + 0.838·91-s − 0.406·97-s + 2.38·101-s + 0.788·103-s + 0.386·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.121592813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.121592813\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 16 T + 134 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 16 T + 134 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.174332855824677635825055184874, −8.798670487828238229396246751454, −8.753377106840359325550546439270, −8.358734294957248580673881981374, −7.77590251642253916492479492068, −7.38079981752865511056317011524, −6.86360148088145715715249238443, −6.70737651959230076982105602655, −6.11065955209596679955876341726, −5.99207301977644215719147173386, −5.27042398139847776916159016083, −4.82662388354012588224526323501, −4.30015104709966765267891163236, −4.28065559316850229570490989414, −3.43559858055096352650808262398, −3.23061528504246566380858646019, −2.30489021656926531144153411702, −1.96916406219888721128211433530, −1.29710336955020146725894352244, −0.69259820703426237827432235039,
0.69259820703426237827432235039, 1.29710336955020146725894352244, 1.96916406219888721128211433530, 2.30489021656926531144153411702, 3.23061528504246566380858646019, 3.43559858055096352650808262398, 4.28065559316850229570490989414, 4.30015104709966765267891163236, 4.82662388354012588224526323501, 5.27042398139847776916159016083, 5.99207301977644215719147173386, 6.11065955209596679955876341726, 6.70737651959230076982105602655, 6.86360148088145715715249238443, 7.38079981752865511056317011524, 7.77590251642253916492479492068, 8.358734294957248580673881981374, 8.753377106840359325550546439270, 8.798670487828238229396246751454, 9.174332855824677635825055184874
