| L(s) = 1 | − 62·7-s − 482·13-s − 542·19-s − 694·25-s − 1.55e3·31-s + 2.15e3·37-s − 596·43-s − 1.91e3·49-s − 5.28e3·61-s + 1.12e4·67-s + 1.43e4·73-s + 658·79-s + 2.98e4·91-s − 3.19e4·97-s + 3.29e3·103-s − 6.47e3·109-s − 1.93e4·121-s + 127-s + 131-s + 3.36e4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.26·7-s − 2.85·13-s − 1.50·19-s − 1.11·25-s − 1.61·31-s + 1.57·37-s − 0.322·43-s − 0.799·49-s − 1.41·61-s + 2.49·67-s + 2.70·73-s + 0.105·79-s + 3.60·91-s − 3.39·97-s + 0.310·103-s − 0.545·109-s − 1.31·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 1.89·133-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.2057850383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2057850383\) |
| \(L(3)\) |
|
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 | $C_2^2$ | \( 1 + 694 T^{2} + p^{8} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 31 T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 19318 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 241 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 118442 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 271 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 511082 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1220162 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 778 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 1079 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 791522 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 298 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 1175638 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6255362 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 16021322 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2641 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5609 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 31383362 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 7199 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 329 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 93167042 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 58951082 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 15961 T + p^{4} T^{2} )^{2} \) |
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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
−13.71823502560815250954889667171, −12.64836593810656607525886313572, −12.45022564739218180290274335378, −11.81010888749555906893882801025, −11.09697444729623078370047166097, −10.58230208507317201608599990142, −9.812538473002807956575916533835, −9.493627489051444843114398073699, −9.375581766525475270988699124412, −8.104689169798971206088529223301, −7.85169865390757920266217397772, −6.90498690418368517750519345952, −6.72334436103308784383730694449, −5.83763632179216992590037879961, −5.13327308327571124320268155979, −4.41114827908719871320249860904, −3.62080101515407406196007772047, −2.62287391120781756648429351322, −2.08148830593690100255799207609, −0.18773472078952524794597144502,
0.18773472078952524794597144502, 2.08148830593690100255799207609, 2.62287391120781756648429351322, 3.62080101515407406196007772047, 4.41114827908719871320249860904, 5.13327308327571124320268155979, 5.83763632179216992590037879961, 6.72334436103308784383730694449, 6.90498690418368517750519345952, 7.85169865390757920266217397772, 8.104689169798971206088529223301, 9.375581766525475270988699124412, 9.493627489051444843114398073699, 9.812538473002807956575916533835, 10.58230208507317201608599990142, 11.09697444729623078370047166097, 11.81010888749555906893882801025, 12.45022564739218180290274335378, 12.64836593810656607525886313572, 13.71823502560815250954889667171
