| L(s) = 1 | − 4·2-s + 2·4-s − 8·7-s + 20·8-s − 2·9-s + 32·14-s − 45·16-s + 8·18-s + 8·19-s + 10·25-s − 16·28-s − 16·32-s − 4·36-s − 32·38-s + 4·41-s + 48·47-s + 30·49-s − 40·50-s − 160·56-s − 16·59-s + 16·63-s + 204·64-s + 8·67-s − 16·71-s − 40·72-s + 16·76-s + 9·81-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4-s − 3.02·7-s + 7.07·8-s − 2/3·9-s + 8.55·14-s − 11.2·16-s + 1.88·18-s + 1.83·19-s + 2·25-s − 3.02·28-s − 2.82·32-s − 2/3·36-s − 5.19·38-s + 0.624·41-s + 7.00·47-s + 30/7·49-s − 5.65·50-s − 21.3·56-s − 2.08·59-s + 2.01·63-s + 51/2·64-s + 0.977·67-s − 1.89·71-s − 4.71·72-s + 1.83·76-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1324198003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1324198003\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 31 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 3 | $C_2^3$ | \( 1 + 2 T^{2} - 5 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 14 T^{2} + 75 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^3$ | \( 1 - 24 T^{2} + 407 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^3$ | \( 1 - 32 T^{2} + 735 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 - 56 T^{2} + 1767 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 14 T^{2} - 1653 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 53 | $C_2^3$ | \( 1 - 88 T^{2} + 4935 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 120 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 8 T - 7 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 - 128 T^{2} + 11055 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^3$ | \( 1 - 30 T^{2} - 5341 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^3$ | \( 1 + 34 T^{2} - 5733 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 128 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
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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
−7.36451520142314166423907149920, −7.09041766574884571405538157913, −7.00769756226760508395376518060, −6.92683328064437950089960531087, −6.28948270897732793321224524487, −5.94003375938243410176721783354, −5.86045083326393903695292025855, −5.64493644231594364733558266637, −5.61014238033827112746463749175, −4.87081849214824631874791745862, −4.86606547180121119284497757437, −4.60955735823945677354041934479, −4.54057362364423006635911837319, −3.88016614558149551521869526144, −3.75862925036686037739637054372, −3.74439204916815026735902377145, −3.20862274106841725068113721339, −3.07489780922602685608747262620, −2.70590754700728615829228563785, −2.21701535235734384552302118984, −1.92079282276790718832722003099, −0.974766130646938283272279433638, −0.953891053201993173704081561013, −0.811796024737047925686806642578, −0.30243993025902016558854143111,
0.30243993025902016558854143111, 0.811796024737047925686806642578, 0.953891053201993173704081561013, 0.974766130646938283272279433638, 1.92079282276790718832722003099, 2.21701535235734384552302118984, 2.70590754700728615829228563785, 3.07489780922602685608747262620, 3.20862274106841725068113721339, 3.74439204916815026735902377145, 3.75862925036686037739637054372, 3.88016614558149551521869526144, 4.54057362364423006635911837319, 4.60955735823945677354041934479, 4.86606547180121119284497757437, 4.87081849214824631874791745862, 5.61014238033827112746463749175, 5.64493644231594364733558266637, 5.86045083326393903695292025855, 5.94003375938243410176721783354, 6.28948270897732793321224524487, 6.92683328064437950089960531087, 7.00769756226760508395376518060, 7.09041766574884571405538157913, 7.36451520142314166423907149920
