| L(s) = 1 | − 57·4-s + 2.22e3·16-s + 1.25e4·25-s − 3.55e4·37-s − 3.36e4·49-s − 6.84e4·64-s − 7.12e5·100-s − 8.78e5·109-s + 6.32e5·121-s + 127-s + 131-s + 137-s + 139-s + 2.02e6·148-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 + 1.91e6·196-s + 197-s + ⋯ |
| L(s) = 1 | − 1.78·4-s + 2.17·16-s + 4·25-s − 4.26·37-s − 2·49-s − 2.08·64-s − 7.12·100-s − 7.08·109-s + 3.92·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 7.60·148-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 + 3.56·196-s + 1.83e−6·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4451522658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4451522658\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2^2$ | \( 1 + 57 T^{2} + p^{10} T^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 316326 T^{2} + p^{10} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 11649618 T^{2} + p^{10} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 12005226 T^{2} + p^{10} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + 8886 T + p^{5} T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - 11748 T + p^{5} T^{2} )^{2}( 1 + 11748 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 233688486 T^{2} + p^{10} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - 69364 T + p^{5} T^{2} )^{2}( 1 + 69364 T + p^{5} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 3603512526 T^{2} + p^{10} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 80168 T + p^{5} T^{2} )^{2}( 1 + 80168 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
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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
−7.86576104185211267215124929145, −7.79982823797305368212606932309, −7.38855804325426625993165293994, −6.91984294366985158964847731525, −6.66523178540713297862540553903, −6.64835806346413960738195019834, −6.61817626320507270623710372814, −5.75408774422612487340177227376, −5.49299260758007060381447405031, −5.39448720416705162704586146623, −5.03710668991899306622088338092, −4.95765659139868467633761291468, −4.51552010529967204594095717836, −4.36195750818827394295673425585, −3.99302120170304300330144317727, −3.50760299121264961327448382857, −3.26379361491310290869403149250, −3.08989431158575005916416305569, −2.81037207950037665839628135275, −2.12401696743252256494628990706, −1.68165184654475193440795629667, −1.31532034130382829486404750094, −1.04143814443509192047046397009, −0.55217882246384970717372664891, −0.12729441656494325706204509780,
0.12729441656494325706204509780, 0.55217882246384970717372664891, 1.04143814443509192047046397009, 1.31532034130382829486404750094, 1.68165184654475193440795629667, 2.12401696743252256494628990706, 2.81037207950037665839628135275, 3.08989431158575005916416305569, 3.26379361491310290869403149250, 3.50760299121264961327448382857, 3.99302120170304300330144317727, 4.36195750818827394295673425585, 4.51552010529967204594095717836, 4.95765659139868467633761291468, 5.03710668991899306622088338092, 5.39448720416705162704586146623, 5.49299260758007060381447405031, 5.75408774422612487340177227376, 6.61817626320507270623710372814, 6.64835806346413960738195019834, 6.66523178540713297862540553903, 6.91984294366985158964847731525, 7.38855804325426625993165293994, 7.79982823797305368212606932309, 7.86576104185211267215124929145
