| L(s) = 1 | − 14·7-s + 28·13-s + 16·19-s − 31·25-s + 70·31-s + 88·37-s − 44·43-s + 49·49-s + 40·61-s + 28·67-s + 178·73-s + 220·79-s − 392·91-s + 22·97-s − 44·103-s − 104·109-s + 161·121-s + 127-s + 131-s − 224·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2·7-s + 2.15·13-s + 0.842·19-s − 1.23·25-s + 2.25·31-s + 2.37·37-s − 1.02·43-s + 49-s + 0.655·61-s + 0.417·67-s + 2.43·73-s + 2.78·79-s − 4.30·91-s + 0.226·97-s − 0.427·103-s − 0.954·109-s + 1.33·121-s + 0.00787·127-s + 0.00763·131-s − 1.68·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.365100568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.365100568\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 31 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 161 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 254 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 238 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1358 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 35 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 44 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2066 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 1502 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5537 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6638 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 20 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 5794 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 89 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 110 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 13049 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15518 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{2} \) |
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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
−13.77755173805365306314427564469, −13.32400762138401490995873481208, −12.77064333225368885393886034155, −12.27763000891051257122473646923, −11.47249956539311320157921856633, −11.30700392146741847634093170779, −10.44805001704607546789308730683, −9.812943568856192976160582953005, −9.626863263157341712288610308841, −9.010188945624815320013604632274, −8.131754965503335801945351869534, −7.903536057022504246604496228115, −6.72210216652094354861510849865, −6.23816933355328191565619285740, −6.16503019899032015256015540590, −5.07357916438979434651728887417, −3.89472861492659457570043490807, −3.51759935779301444893503367775, −2.64442532676650251209430376905, −0.920305415319379111537155552069,
0.920305415319379111537155552069, 2.64442532676650251209430376905, 3.51759935779301444893503367775, 3.89472861492659457570043490807, 5.07357916438979434651728887417, 6.16503019899032015256015540590, 6.23816933355328191565619285740, 6.72210216652094354861510849865, 7.903536057022504246604496228115, 8.131754965503335801945351869534, 9.010188945624815320013604632274, 9.626863263157341712288610308841, 9.812943568856192976160582953005, 10.44805001704607546789308730683, 11.30700392146741847634093170779, 11.47249956539311320157921856633, 12.27763000891051257122473646923, 12.77064333225368885393886034155, 13.32400762138401490995873481208, 13.77755173805365306314427564469
