| L(s) = 1 | + 14·7-s + 96·19-s + 13·25-s + 66·31-s − 32·37-s − 80·43-s + 49·49-s + 84·61-s − 160·67-s − 24·73-s + 242·79-s + 72·103-s + 160·109-s + 53·121-s + 127-s + 131-s + 1.34e3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 460·169-s + 173-s + 182·175-s + ⋯ |
| L(s) = 1 | + 2·7-s + 5.05·19-s + 0.519·25-s + 2.12·31-s − 0.864·37-s − 1.86·43-s + 49-s + 1.37·61-s − 2.38·67-s − 0.328·73-s + 3.06·79-s + 0.699·103-s + 1.46·109-s + 0.438·121-s + 0.00787·127-s + 0.00763·131-s + 10.1·133-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 + 2.72·169-s + 0.00578·173-s + 1.03·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(6.179894967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.179894967\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 - 13 T^{2} - 456 T^{4} - 13 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 53 T^{2} - 11832 T^{4} - 53 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 230 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 430 T^{2} + 101379 T^{4} - 430 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 37 T + p^{2} T^{2} )^{2}( 1 - 11 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 302 T^{2} - 188637 T^{4} - 302 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 19 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )^{2}( 1 + 13 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 16 T - 1113 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 3110 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 20 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 + 4166 T^{2} + 12475875 T^{4} + 4166 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 893 T^{2} - 7093032 T^{4} - 893 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2^3$ | \( 1 + 6899 T^{2} + 35478840 T^{4} + 6899 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 42 T + 4309 T^{2} - 42 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 80 T + 1911 T^{2} + 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 9326 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 12 T + 5377 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 121 T + 8400 T^{2} - 121 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 13715 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 + 13574 T^{2} + 121511235 T^{4} + 13574 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 6025 T^{2} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.396577615470032018405643616381, −8.270706351012854170804515615187, −7.85230732440508368112134875464, −7.77504397823712355552566019995, −7.67517248065706782900271371886, −7.41237370292518868869796173391, −6.91652930916056959835342820948, −6.71823311321117016655808760912, −6.61708439757059604085198644338, −6.02159467744739230736597143999, −5.61149695323681853684503604043, −5.52030118339935836722391735215, −5.17638791662003148577863365622, −4.89474677232061233583305324962, −4.74992420303077815426741765599, −4.63855034962747979710118455708, −3.91083895925383236401633058287, −3.52000524310423555451113398632, −3.29689052451462220036175916199, −2.95725045680650150232312291236, −2.61207577459861082066547302482, −1.91544000788752737975929060996, −1.47220006792357928828671353740, −1.14383368409811486878071503821, −0.76070325850803850832547776759,
0.76070325850803850832547776759, 1.14383368409811486878071503821, 1.47220006792357928828671353740, 1.91544000788752737975929060996, 2.61207577459861082066547302482, 2.95725045680650150232312291236, 3.29689052451462220036175916199, 3.52000524310423555451113398632, 3.91083895925383236401633058287, 4.63855034962747979710118455708, 4.74992420303077815426741765599, 4.89474677232061233583305324962, 5.17638791662003148577863365622, 5.52030118339935836722391735215, 5.61149695323681853684503604043, 6.02159467744739230736597143999, 6.61708439757059604085198644338, 6.71823311321117016655808760912, 6.91652930916056959835342820948, 7.41237370292518868869796173391, 7.67517248065706782900271371886, 7.77504397823712355552566019995, 7.85230732440508368112134875464, 8.270706351012854170804515615187, 8.396577615470032018405643616381
