| L(s) = 1 | − 2·4-s − 3·9-s + 2·11-s + 7·16-s − 12·19-s − 36·29-s + 12·31-s + 6·36-s + 2·41-s − 4·44-s + 9·49-s + 16·59-s + 18·61-s − 4·64-s − 22·71-s + 24·76-s − 10·79-s + 6·81-s − 22·89-s − 6·99-s + 16·101-s − 32·109-s + 72·116-s − 31·121-s − 24·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 4-s − 9-s + 0.603·11-s + 7/4·16-s − 2.75·19-s − 6.68·29-s + 2.15·31-s + 36-s + 0.312·41-s − 0.603·44-s + 9/7·49-s + 2.08·59-s + 2.30·61-s − 1/2·64-s − 2.61·71-s + 2.75·76-s − 1.12·79-s + 2/3·81-s − 2.33·89-s − 0.603·99-s + 1.59·101-s − 3.06·109-s + 6.68·116-s − 2.81·121-s − 2.15·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05764482612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05764482612\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 \) |
| 13 | \( ( 1 + T^{2} )^{3} \) |
| good | 2 | \( ( 1 - p T + 3 T^{2} - 3 p T^{3} + 3 p T^{4} - p^{3} T^{5} + p^{3} T^{6} )( 1 + p T + 3 T^{2} + 3 p T^{3} + 3 p T^{4} + p^{3} T^{5} + p^{3} T^{6} ) \) |
| 7 | \( 1 - 9 T^{2} + 5 p T^{4} - 38 T^{6} + 5 p^{3} T^{8} - 9 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 - T + 17 T^{2} - 38 T^{3} + 17 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 - 37 T^{2} + 1091 T^{4} - 19758 T^{6} + 1091 p^{2} T^{8} - 37 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 + 6 T + 41 T^{2} + 164 T^{3} + 41 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 57 T^{2} + 2531 T^{4} - 64070 T^{6} + 2531 p^{2} T^{8} - 57 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 + 6 T + p T^{2} )^{6} \) |
| 31 | \( ( 1 - 6 T + 77 T^{2} - 340 T^{3} + 77 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 53 T^{2} + 3739 T^{4} - 133022 T^{6} + 3739 p^{2} T^{8} - 53 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - T + 91 T^{2} - 6 T^{3} + 91 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 34 T^{2} + 1751 T^{4} - 167484 T^{6} + 1751 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 118 T^{2} + 6399 T^{4} - 283732 T^{6} + 6399 p^{2} T^{8} - 118 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 213 T^{2} + 20027 T^{4} - 1223966 T^{6} + 20027 p^{2} T^{8} - 213 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 8 T + 129 T^{2} - 816 T^{3} + 129 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 9 T + 71 T^{2} - 254 T^{3} + 71 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 258 T^{2} + 33863 T^{4} - 2806460 T^{6} + 33863 p^{2} T^{8} - 258 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 11 T + 237 T^{2} + 1530 T^{3} + 237 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 202 T^{2} + 25151 T^{4} - 2178828 T^{6} + 25151 p^{2} T^{8} - 202 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 5 T + 189 T^{2} + 854 T^{3} + 189 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 338 T^{2} + 54567 T^{4} - 5528348 T^{6} + 54567 p^{2} T^{8} - 338 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 11 T + 275 T^{2} + 1954 T^{3} + 275 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 309 T^{2} + 53987 T^{4} - 6424526 T^{6} + 53987 p^{2} T^{8} - 309 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.24675698141683186756169204556, −5.23827549455576444617684915849, −5.08393976262261890803366787068, −5.04415825064435219545266126214, −4.33745900979908265966801562894, −4.30039198709771846893514865854, −4.29599704310334540860921140291, −4.26732705592754898469084346296, −3.88456014434641877908589326214, −3.85096843753709715796561504672, −3.66222883397992151075454980071, −3.64011536411592464163751034647, −3.42225285713798905431475253852, −2.96467997143573872064997596684, −2.76368165776181243297415920056, −2.74589052912346347466346002477, −2.42298471586196053943453682593, −2.24072169709271762784196079828, −2.06496727807127902582103824105, −1.71510952892384484279799066133, −1.64750551038587228950260744757, −1.23611321405951556586516670284, −1.02974877031961659798242959458, −0.46982835383278973730882110018, −0.05889724693817678428696016877,
0.05889724693817678428696016877, 0.46982835383278973730882110018, 1.02974877031961659798242959458, 1.23611321405951556586516670284, 1.64750551038587228950260744757, 1.71510952892384484279799066133, 2.06496727807127902582103824105, 2.24072169709271762784196079828, 2.42298471586196053943453682593, 2.74589052912346347466346002477, 2.76368165776181243297415920056, 2.96467997143573872064997596684, 3.42225285713798905431475253852, 3.64011536411592464163751034647, 3.66222883397992151075454980071, 3.85096843753709715796561504672, 3.88456014434641877908589326214, 4.26732705592754898469084346296, 4.29599704310334540860921140291, 4.30039198709771846893514865854, 4.33745900979908265966801562894, 5.04415825064435219545266126214, 5.08393976262261890803366787068, 5.23827549455576444617684915849, 5.24675698141683186756169204556
Plot not available for L-functions of degree greater than 10.
