| L(s) = 1 | − 2·2-s + 3·4-s + 2·5-s − 2·7-s − 4·8-s − 4·10-s + 4·11-s − 5·13-s + 4·14-s + 5·16-s + 7·17-s − 5·19-s + 6·20-s − 8·22-s − 2·23-s + 3·25-s + 10·26-s − 6·28-s − 4·29-s − 12·31-s − 6·32-s − 14·34-s − 4·35-s + 5·37-s + 10·38-s − 8·40-s + 4·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.894·5-s − 0.755·7-s − 1.41·8-s − 1.26·10-s + 1.20·11-s − 1.38·13-s + 1.06·14-s + 5/4·16-s + 1.69·17-s − 1.14·19-s + 1.34·20-s − 1.70·22-s − 0.417·23-s + 3/5·25-s + 1.96·26-s − 1.13·28-s − 0.742·29-s − 2.15·31-s − 1.06·32-s − 2.40·34-s − 0.676·35-s + 0.821·37-s + 1.62·38-s − 1.26·40-s + 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76212900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76212900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.538028648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.538028648\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 97 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 31 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 7 T + 45 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 43 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 79 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 27 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 89 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 5 T + 113 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 151 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 14 T + 178 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 162 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + T + 177 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−7.80004876829237239821160220492, −7.64960453625031562917977428818, −7.18621833721914412862987981430, −7.16845861227911437232860782994, −6.49343013675081358706075997260, −6.42818307990633955742424085811, −5.87280566947661063522452313570, −5.82371701983833783251030442049, −5.24923031547997611067407930841, −5.04176849484867455987142011239, −4.31089038846165333043060993378, −3.94682925994524226127180017726, −3.49791559992987913803443446681, −3.27606045721848197830866789816, −2.49415151120311090313334858751, −2.40203852255559406094112833524, −1.81301558920480399911692576218, −1.59490917075044279055796442823, −0.791319684248509900258007121403, −0.46515419168380556438610880824,
0.46515419168380556438610880824, 0.791319684248509900258007121403, 1.59490917075044279055796442823, 1.81301558920480399911692576218, 2.40203852255559406094112833524, 2.49415151120311090313334858751, 3.27606045721848197830866789816, 3.49791559992987913803443446681, 3.94682925994524226127180017726, 4.31089038846165333043060993378, 5.04176849484867455987142011239, 5.24923031547997611067407930841, 5.82371701983833783251030442049, 5.87280566947661063522452313570, 6.42818307990633955742424085811, 6.49343013675081358706075997260, 7.16845861227911437232860782994, 7.18621833721914412862987981430, 7.64960453625031562917977428818, 7.80004876829237239821160220492
