| L(s) = 1 | − 2·2-s + 3·4-s + 2·7-s − 4·8-s + 6·11-s − 10·13-s − 4·14-s + 5·16-s − 2·19-s − 12·22-s + 20·26-s + 6·28-s + 6·29-s − 8·31-s − 6·32-s + 2·37-s + 4·38-s + 12·41-s − 4·43-s + 18·44-s + 12·47-s + 49-s − 30·52-s + 6·53-s − 8·56-s − 12·58-s + 18·59-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.755·7-s − 1.41·8-s + 1.80·11-s − 2.77·13-s − 1.06·14-s + 5/4·16-s − 0.458·19-s − 2.55·22-s + 3.92·26-s + 1.13·28-s + 1.11·29-s − 1.43·31-s − 1.06·32-s + 0.328·37-s + 0.648·38-s + 1.87·41-s − 0.609·43-s + 2.71·44-s + 1.75·47-s + 1/7·49-s − 4.16·52-s + 0.824·53-s − 1.06·56-s − 1.57·58-s + 2.34·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.500452786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.500452786\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 10 T + 48 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 64 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 48 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 127 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 112 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 18 T + 196 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 108 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 18 T + 172 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 187 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 147 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.585961035340322910504969741762, −8.564351939549586200275490031057, −7.63378963683121853855758676117, −7.60553535900320545667296771853, −7.24133422572765208007956838028, −7.16243791595589116813123827045, −6.46886008021186023190180983011, −6.26717189261945367055548223165, −5.77997415909167863345592406906, −5.31642141141855173206294673717, −4.79132564062844985577578188614, −4.59348328407234204001486845554, −3.92196839120839761454897008847, −3.74738704736386788635380242209, −2.77204138034596061285246910137, −2.62505839480781160542173399591, −1.96826623906596701698077805252, −1.81182661570116069200964044476, −0.899992437945576909093392677124, −0.55819194998023650950123426304,
0.55819194998023650950123426304, 0.899992437945576909093392677124, 1.81182661570116069200964044476, 1.96826623906596701698077805252, 2.62505839480781160542173399591, 2.77204138034596061285246910137, 3.74738704736386788635380242209, 3.92196839120839761454897008847, 4.59348328407234204001486845554, 4.79132564062844985577578188614, 5.31642141141855173206294673717, 5.77997415909167863345592406906, 6.26717189261945367055548223165, 6.46886008021186023190180983011, 7.16243791595589116813123827045, 7.24133422572765208007956838028, 7.60553535900320545667296771853, 7.63378963683121853855758676117, 8.564351939549586200275490031057, 8.585961035340322910504969741762
