| L(s) = 1 | − 4·3-s + 8·7-s + 6·9-s + 12·13-s − 16-s − 32·21-s + 4·27-s − 28·31-s + 12·37-s − 48·39-s + 4·48-s + 32·49-s − 40·61-s + 48·63-s + 12·67-s − 37·81-s + 96·91-s + 112·93-s + 32·97-s + 48·109-s − 48·111-s − 8·112-s + 72·117-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 3.02·7-s + 2·9-s + 3.32·13-s − 1/4·16-s − 6.98·21-s + 0.769·27-s − 5.02·31-s + 1.97·37-s − 7.68·39-s + 0.577·48-s + 32/7·49-s − 5.12·61-s + 6.04·63-s + 1.46·67-s − 4.11·81-s + 10.0·91-s + 11.6·93-s + 3.24·97-s + 4.59·109-s − 4.55·111-s − 0.755·112-s + 6.65·117-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.462431052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.462431052\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 + 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 + 2722 T^{4} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 1054 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 6286 T^{4} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^3$ | \( 1 + 5794 T^{4} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 - 13294 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 160 T^{2} + p^{2} T^{4} )( 1 + 160 T^{2} + p^{2} T^{4} ) \) |
| 97 | $C_2^2$ | \( ( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.151527824217868199772553082084, −7.985667336992610258544414068342, −7.55815401452421260589109160576, −7.55257768606524366706760087608, −7.31448936927655334657759213355, −6.91077959633222294199376639543, −6.49977554905428867559751523115, −6.23390578994260495613547511873, −6.12902095355161620761522031492, −5.78129055427179953965725591339, −5.66861025905782657992801058871, −5.52988762381361528154090112068, −5.15916527899641744449944928843, −4.84344528957901258815161864057, −4.64259234660912567148835955648, −4.32237017230639587145641179790, −4.23245917246151748589438039087, −3.61449116945841644620462352411, −3.35581357064405588198968368215, −3.18681555762922422595595509476, −2.11602835823495382303226409215, −1.97109278796746556937353066664, −1.48187566224242678566983793566, −1.28122852765509102806085250010, −0.62798853802220257887368018721,
0.62798853802220257887368018721, 1.28122852765509102806085250010, 1.48187566224242678566983793566, 1.97109278796746556937353066664, 2.11602835823495382303226409215, 3.18681555762922422595595509476, 3.35581357064405588198968368215, 3.61449116945841644620462352411, 4.23245917246151748589438039087, 4.32237017230639587145641179790, 4.64259234660912567148835955648, 4.84344528957901258815161864057, 5.15916527899641744449944928843, 5.52988762381361528154090112068, 5.66861025905782657992801058871, 5.78129055427179953965725591339, 6.12902095355161620761522031492, 6.23390578994260495613547511873, 6.49977554905428867559751523115, 6.91077959633222294199376639543, 7.31448936927655334657759213355, 7.55257768606524366706760087608, 7.55815401452421260589109160576, 7.985667336992610258544414068342, 8.151527824217868199772553082084
