| L(s) = 1 | + 2·2-s − 4·3-s + 3·4-s − 2·5-s − 8·6-s + 4·8-s + 6·9-s − 4·10-s − 12·12-s + 2·13-s + 8·15-s + 5·16-s + 2·17-s + 12·18-s − 6·20-s − 4·23-s − 16·24-s − 4·25-s + 4·26-s + 4·27-s − 2·29-s + 16·30-s − 6·31-s + 6·32-s + 4·34-s + 18·36-s − 8·37-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 2.30·3-s + 3/2·4-s − 0.894·5-s − 3.26·6-s + 1.41·8-s + 2·9-s − 1.26·10-s − 3.46·12-s + 0.554·13-s + 2.06·15-s + 5/4·16-s + 0.485·17-s + 2.82·18-s − 1.34·20-s − 0.834·23-s − 3.26·24-s − 4/5·25-s + 0.784·26-s + 0.769·27-s − 0.371·29-s + 2.92·30-s − 1.07·31-s + 1.06·32-s + 0.685·34-s + 3·36-s − 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8076964 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8076964 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 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 | $D_{4}$ | \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 44 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 62 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 18 T + 196 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 98 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 14 T + 192 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T - 76 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 22 T + 272 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 216 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−8.384317641005347752864054160804, −8.048820929375021945754628518472, −7.58505852444811383056955677834, −7.35899531521778531502122919835, −6.83403409000343597068862487034, −6.47987805899750528826487442809, −6.09248673828980813214726943106, −5.81868690948606170480661028541, −5.48824965809634287120661905165, −5.33625844595944660158278373400, −4.68464308557591876629991324699, −4.36827935626596524989611145236, −4.03917110304674397975380623077, −3.55475410255333249863939777162, −3.07726971834435539859597113764, −2.56538632160654225482998505339, −1.59546814137714372138663395777, −1.29922312819189031317424962441, 0, 0,
1.29922312819189031317424962441, 1.59546814137714372138663395777, 2.56538632160654225482998505339, 3.07726971834435539859597113764, 3.55475410255333249863939777162, 4.03917110304674397975380623077, 4.36827935626596524989611145236, 4.68464308557591876629991324699, 5.33625844595944660158278373400, 5.48824965809634287120661905165, 5.81868690948606170480661028541, 6.09248673828980813214726943106, 6.47987805899750528826487442809, 6.83403409000343597068862487034, 7.35899531521778531502122919835, 7.58505852444811383056955677834, 8.048820929375021945754628518472, 8.384317641005347752864054160804
