| L(s) = 1 | + 4·3-s + 4-s + 4·9-s + 12·11-s + 4·12-s + 4·13-s + 4·16-s + 12·17-s + 8·19-s − 6·23-s + 7·25-s − 4·27-s + 6·29-s + 2·31-s + 48·33-s + 4·36-s + 14·37-s + 16·39-s − 10·43-s + 12·44-s − 12·47-s + 16·48-s + 48·51-s + 4·52-s + 6·53-s + 32·57-s − 18·59-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1/2·4-s + 4/3·9-s + 3.61·11-s + 1.15·12-s + 1.10·13-s + 16-s + 2.91·17-s + 1.83·19-s − 1.25·23-s + 7/5·25-s − 0.769·27-s + 1.11·29-s + 0.359·31-s + 8.35·33-s + 2/3·36-s + 2.30·37-s + 2.56·39-s − 1.52·43-s + 1.80·44-s − 1.75·47-s + 2.30·48-s + 6.72·51-s + 0.554·52-s + 0.824·53-s + 4.23·57-s − 2.34·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.27297583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.27297583\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_2^3$ | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | $D_{4}$ | \( ( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^3$ | \( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 12 T + 77 T^{2} - 396 T^{3} + 1752 T^{4} - 396 p T^{5} + 77 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T - 16 T^{2} + 36 T^{3} + 1347 T^{4} + 36 p T^{5} - 16 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2 T - 32 T^{2} + 52 T^{3} + 211 T^{4} + 52 p T^{5} - 32 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 - 55 T^{2} + 1344 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 10 T + 16 T^{2} - 20 T^{3} + 907 T^{4} - 20 p T^{5} + 16 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 12 T + 62 T^{2} - 144 T^{3} - 2253 T^{4} - 144 p T^{5} + 62 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 6 T - 31 T^{2} + 234 T^{3} - 228 T^{4} + 234 p T^{5} - 31 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 18 T + 128 T^{2} + 1404 T^{3} + 15819 T^{4} + 1404 p T^{5} + 128 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 20 T + 195 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 2 T + 108 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 4 T - 107 T^{2} - 92 T^{3} + 8632 T^{4} - 92 p T^{5} - 107 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 22 T + 232 T^{2} + 2068 T^{3} + 19027 T^{4} + 2068 p T^{5} + 232 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T - 22 T^{2} - 144 T^{3} + 6819 T^{4} - 144 p T^{5} - 22 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 8 T - 38 T^{2} + 736 T^{3} - 5213 T^{4} + 736 p T^{5} - 38 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.80677294083538971185625046011, −7.68789135010208506338016463596, −7.26697016185631552067609119228, −6.82225821130460254350701127879, −6.65187018178371575207458865701, −6.59689423671353032571441097843, −6.07727397981684393289655126780, −6.01442183063829316089581830239, −5.96504570216047742131377617266, −5.55395417790959195666589958763, −5.26163340426782698127095744489, −4.69345671031228386476434185189, −4.67621527224021222821833072735, −4.16689541323760932524433080054, −3.94774941028514516189247652443, −3.72821997196758380704602672391, −3.27751636068378832141753391100, −3.17733732156672023115378016073, −3.11465916460570657844776950902, −2.72888216077880689284527216888, −2.72737664519015519976383417806, −1.52902241141656858176993947104, −1.51588341035377775840233055427, −1.26131956013346689223843020942, −1.18908838177968779876003625229,
1.18908838177968779876003625229, 1.26131956013346689223843020942, 1.51588341035377775840233055427, 1.52902241141656858176993947104, 2.72737664519015519976383417806, 2.72888216077880689284527216888, 3.11465916460570657844776950902, 3.17733732156672023115378016073, 3.27751636068378832141753391100, 3.72821997196758380704602672391, 3.94774941028514516189247652443, 4.16689541323760932524433080054, 4.67621527224021222821833072735, 4.69345671031228386476434185189, 5.26163340426782698127095744489, 5.55395417790959195666589958763, 5.96504570216047742131377617266, 6.01442183063829316089581830239, 6.07727397981684393289655126780, 6.59689423671353032571441097843, 6.65187018178371575207458865701, 6.82225821130460254350701127879, 7.26697016185631552067609119228, 7.68789135010208506338016463596, 7.80677294083538971185625046011
