| L(s) = 1 | − 362·9-s − 200·11-s − 4.48e3·19-s − 1.57e4·29-s − 4.28e3·31-s − 1.48e4·41-s − 1.86e4·49-s + 5.19e4·59-s − 6.11e3·61-s + 7.52e4·71-s + 1.59e5·79-s + 7.19e4·81-s − 1.65e3·89-s + 7.24e4·99-s − 2.87e5·101-s − 2.12e5·109-s − 2.92e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.02e5·169-s + ⋯ |
| L(s) = 1 | − 1.48·9-s − 0.498·11-s − 2.85·19-s − 3.46·29-s − 0.801·31-s − 1.37·41-s − 1.10·49-s + 1.94·59-s − 0.210·61-s + 1.77·71-s + 2.87·79-s + 1.21·81-s − 0.0221·89-s + 0.742·99-s − 2.80·101-s − 1.71·109-s − 1.81·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s − 0.545·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 362 T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 18610 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 100 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 202442 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 1879458 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2244 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 1185810 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 7854 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2144 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 30515770 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 7414 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 21442214 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 369731298 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 248174170 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 25972 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 3058 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 755362070 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 37608 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 3569749522 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 79728 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 7612675530 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 826 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 15761405890 T^{2} + p^{10} T^{4} \) |
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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
−12.67585835205348999170854419900, −12.24330839913697054327809927871, −11.33064948667372842539259205206, −11.11410167348294412722254943147, −10.74134172529631037994590362221, −9.995853336418572822044772634865, −9.157101274099956349163286097381, −8.933534642757466502257486045098, −8.022726134370093879950639880137, −7.987305189432878214990320224044, −6.78213775220875348897740527142, −6.42421251929629964659484363145, −5.42811791839063485004856778577, −5.32320978893713176414285399260, −4.04052364108158518873225535895, −3.54702864504351796837508709345, −2.38987156984344766018748540915, −1.91387295250370407848645158679, 0, 0,
1.91387295250370407848645158679, 2.38987156984344766018748540915, 3.54702864504351796837508709345, 4.04052364108158518873225535895, 5.32320978893713176414285399260, 5.42811791839063485004856778577, 6.42421251929629964659484363145, 6.78213775220875348897740527142, 7.987305189432878214990320224044, 8.022726134370093879950639880137, 8.933534642757466502257486045098, 9.157101274099956349163286097381, 9.995853336418572822044772634865, 10.74134172529631037994590362221, 11.11410167348294412722254943147, 11.33064948667372842539259205206, 12.24330839913697054327809927871, 12.67585835205348999170854419900
