| L(s) = 1 | + 1.25e4·25-s + 5.57e4·41-s − 6.72e4·49-s − 1.14e5·81-s + 1.02e6·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.48e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 4·25-s + 5.17·41-s − 4·49-s − 1.93·81-s + 7.55·113-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 4·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + 1.83e−6·197-s + 1.79e−6·199-s + 1.54e−6·211-s + 1.34e−6·223-s + 1.28e−6·227-s + 1.26e−6·229-s + 1.20e−6·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(8.333529183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.333529183\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 44 T + 968 T^{2} - 44 p^{5} T^{3} + p^{10} T^{4} )( 1 + 44 T + 968 T^{2} + 44 p^{5} T^{3} + p^{10} T^{4} ) \) |
| 5 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 916 T + 419528 T^{2} - 916 p^{5} T^{3} + p^{10} T^{4} )( 1 + 916 T + 419528 T^{2} + 916 p^{5} T^{3} + p^{10} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 823682 T^{2} + p^{10} T^{4} )^{2} \) |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 1788 T + 1598472 T^{2} - 1788 p^{5} T^{3} + p^{10} T^{4} )( 1 + 1788 T + 1598472 T^{2} + 1788 p^{5} T^{3} + p^{10} T^{4} ) \) |
| 23 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - 13926 T + p^{5} T^{2} )^{4} \) |
| 43 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 12612 T + 79531272 T^{2} - 12612 p^{5} T^{3} + p^{10} T^{4} )( 1 + 12612 T + 79531272 T^{2} + 12612 p^{5} T^{3} + p^{10} T^{4} ) \) |
| 47 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 59 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 31900 T + 508805000 T^{2} - 31900 p^{5} T^{3} + p^{10} T^{4} )( 1 + 31900 T + 508805000 T^{2} + 31900 p^{5} T^{3} + p^{10} T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 42108 T + 886541832 T^{2} - 42108 p^{5} T^{3} + p^{10} T^{4} )( 1 + 42108 T + 886541832 T^{2} + 42108 p^{5} T^{3} + p^{10} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 1605781582 T^{2} + p^{10} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 124756 T + 7782029768 T^{2} - 124756 p^{5} T^{3} + p^{10} T^{4} )( 1 + 124756 T + 7782029768 T^{2} + 124756 p^{5} T^{3} + p^{10} T^{4} ) \) |
| 89 | $C_2^2$ | \( ( 1 - 11116019374 T^{2} + p^{10} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 9872978014 T^{2} + p^{10} T^{4} )^{2} \) |
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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
−7.00260091942284801491974941039, −6.87995433446874067467533809971, −6.45853992500349321573314553789, −6.35850670875180197541580115644, −6.12453845067219281046908657115, −5.84060453157365513151748012179, −5.65160158347975274613187597631, −5.22229871319250226668330640011, −4.97679026397971916028170279227, −4.72301199207540781069907851532, −4.58196801557552557968015454863, −4.39278917455466311360304913245, −4.07108062413186912133183561466, −3.54733019236351729521637333141, −3.44071438293388032634230188261, −2.97029868160968896056834337653, −2.88885240396232243081376586935, −2.59535948963148131608180243742, −2.33095171060522077067334454445, −1.86387462094873831515541472308, −1.44343357890327619644749456322, −1.17014001041901337140585810077, −0.939705153860309124071436389708, −0.50668142762773924761373422975, −0.39336402376564924478872184495,
0.39336402376564924478872184495, 0.50668142762773924761373422975, 0.939705153860309124071436389708, 1.17014001041901337140585810077, 1.44343357890327619644749456322, 1.86387462094873831515541472308, 2.33095171060522077067334454445, 2.59535948963148131608180243742, 2.88885240396232243081376586935, 2.97029868160968896056834337653, 3.44071438293388032634230188261, 3.54733019236351729521637333141, 4.07108062413186912133183561466, 4.39278917455466311360304913245, 4.58196801557552557968015454863, 4.72301199207540781069907851532, 4.97679026397971916028170279227, 5.22229871319250226668330640011, 5.65160158347975274613187597631, 5.84060453157365513151748012179, 6.12453845067219281046908657115, 6.35850670875180197541580115644, 6.45853992500349321573314553789, 6.87995433446874067467533809971, 7.00260091942284801491974941039
