| L(s) = 1 | + 6·2-s + 18·4-s − 10·7-s + 36·8-s + 9-s + 2·11-s − 60·14-s + 52·16-s − 6·17-s + 6·18-s + 2·19-s + 12·22-s − 6·23-s − 180·28-s − 4·29-s − 6·31-s + 48·32-s − 36·34-s + 18·36-s + 18·37-s + 12·38-s + 4·41-s + 36·44-s − 36·46-s + 61·49-s − 36·53-s − 360·56-s + ⋯ |
| L(s) = 1 | + 4.24·2-s + 9·4-s − 3.77·7-s + 12.7·8-s + 1/3·9-s + 0.603·11-s − 16.0·14-s + 13·16-s − 1.45·17-s + 1.41·18-s + 0.458·19-s + 2.55·22-s − 1.25·23-s − 34.0·28-s − 0.742·29-s − 1.07·31-s + 8.48·32-s − 6.17·34-s + 3·36-s + 2.95·37-s + 1.94·38-s + 0.624·41-s + 5.42·44-s − 5.30·46-s + 61/7·49-s − 4.94·53-s − 48.1·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.752999167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.752999167\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 4 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 14 T^{2} + 15 p T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 6 T + 24 T^{2} + 72 T^{3} + 59 T^{4} + 72 p T^{5} + 24 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 2 T - 23 T^{2} + 22 T^{3} + 292 T^{4} + 22 p T^{5} - 23 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T + 52 T^{2} + 240 T^{3} + 1347 T^{4} + 240 p T^{5} + 52 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 6 T - 23 T^{2} - 18 T^{3} + 1404 T^{4} - 18 p T^{5} - 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 18 T + 205 T^{2} - 1746 T^{3} + 12036 T^{4} - 1746 p T^{5} + 205 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 2 T + 80 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 110 T^{2} + 6291 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 90 T^{2} + 5891 T^{4} + 90 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 36 T + 642 T^{2} + 7560 T^{3} + 64187 T^{4} + 7560 p T^{5} + 642 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 10 T - 16 T^{2} + 20 T^{3} + 4075 T^{4} + 20 p T^{5} - 16 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 30 T + 473 T^{2} + 5190 T^{3} + 45540 T^{4} + 5190 p T^{5} + 473 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 2 T + 116 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 30 T + 505 T^{2} + 6150 T^{3} + 58596 T^{4} + 6150 p T^{5} + 505 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 6 T - 23 T^{2} + 594 T^{3} - 5604 T^{4} + 594 p T^{5} - 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 8586 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 6 T - 4 T^{2} + 828 T^{3} - 9525 T^{4} + 828 p T^{5} - 4 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 13254 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.73604954991511620515986225357, −7.21585775187993724732727289413, −7.15716781910576466565254516829, −6.81042467943676085693815140863, −6.54787758089028719087239079926, −6.38997482610495799174715210156, −6.33709539773140678720821517948, −5.80443953973680155959918655087, −5.79357867819380218714671732897, −5.78489045117012556448294224486, −5.60136736160809094786064007969, −4.77102180321959370430531603636, −4.59915379555740688556182183768, −4.50033114044562373292835902658, −4.42894267632393842110637277179, −3.97774006766146059012626886139, −3.72262554755133066376426886682, −3.60187576032734780468868869338, −3.26795725673336442316719879096, −3.01221117458627481195230185276, −2.70653786707697225791758963890, −2.68983911703165740175351440803, −1.91278886851443463868453784016, −1.66473060095041804583089927406, −0.32527822570076160909663539011,
0.32527822570076160909663539011, 1.66473060095041804583089927406, 1.91278886851443463868453784016, 2.68983911703165740175351440803, 2.70653786707697225791758963890, 3.01221117458627481195230185276, 3.26795725673336442316719879096, 3.60187576032734780468868869338, 3.72262554755133066376426886682, 3.97774006766146059012626886139, 4.42894267632393842110637277179, 4.50033114044562373292835902658, 4.59915379555740688556182183768, 4.77102180321959370430531603636, 5.60136736160809094786064007969, 5.78489045117012556448294224486, 5.79357867819380218714671732897, 5.80443953973680155959918655087, 6.33709539773140678720821517948, 6.38997482610495799174715210156, 6.54787758089028719087239079926, 6.81042467943676085693815140863, 7.15716781910576466565254516829, 7.21585775187993724732727289413, 7.73604954991511620515986225357
