| L(s) = 1 | − 16·9-s − 56·11-s − 40·25-s + 56·29-s + 62·49-s + 64·71-s + 304·79-s + 30·81-s + 896·99-s + 296·109-s + 1.47e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 656·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 1.77·9-s − 5.09·11-s − 8/5·25-s + 1.93·29-s + 1.26·49-s + 0.901·71-s + 3.84·79-s + 0.370·81-s + 9.05·99-s + 2.71·109-s + 12.1·121-s + 0.00787·127-s + 0.00763·131-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 − 3.88·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7026461034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7026461034\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 8 p T^{2} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 62 T^{2} + p^{4} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + 328 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 538 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 88 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 914 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 482 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2414 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 3002 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 1934 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 2458 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 2702 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 6872 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 3032 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 1426 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 6658 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 76 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 8488 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 12602 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 13978 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−7.72902976866047530754817485991, −7.52084930229740584925511925029, −7.22063476459390010492384443244, −6.71090933668969609676393932667, −6.64814043700421960211865788874, −6.28955088966645892753930971211, −5.87991185812079386102938552216, −5.69628117945090452690160273232, −5.58849580934222708569091467168, −5.48470801740968221709066742283, −5.02538708724353997295917212812, −4.89892376547895950853688563483, −4.73133064143032648434073841849, −4.47892317734127450559411228053, −3.80216004714425403692442460676, −3.64338172017229079781172550503, −3.19792585246473343816506542262, −2.93375486564566336744836396032, −2.78676732959930835206095969368, −2.29815274893213798139575876993, −2.21176869459850239082712209521, −2.20940472568138533451542597377, −1.10620557036071347937412868380, −0.44252860729259933139155984956, −0.29513670408798768302587522819,
0.29513670408798768302587522819, 0.44252860729259933139155984956, 1.10620557036071347937412868380, 2.20940472568138533451542597377, 2.21176869459850239082712209521, 2.29815274893213798139575876993, 2.78676732959930835206095969368, 2.93375486564566336744836396032, 3.19792585246473343816506542262, 3.64338172017229079781172550503, 3.80216004714425403692442460676, 4.47892317734127450559411228053, 4.73133064143032648434073841849, 4.89892376547895950853688563483, 5.02538708724353997295917212812, 5.48470801740968221709066742283, 5.58849580934222708569091467168, 5.69628117945090452690160273232, 5.87991185812079386102938552216, 6.28955088966645892753930971211, 6.64814043700421960211865788874, 6.71090933668969609676393932667, 7.22063476459390010492384443244, 7.52084930229740584925511925029, 7.72902976866047530754817485991
