| L(s) = 1 | − 54·9-s + 352·13-s + 432·17-s + 250·25-s − 2.16e3·29-s − 1.07e3·37-s + 2.18e3·41-s + 4.66e3·49-s + 4.70e3·53-s − 1.00e3·61-s − 1.32e4·73-s + 2.18e3·81-s − 3.45e4·89-s + 1.22e4·97-s + 3.89e4·101-s − 4.61e3·109-s − 8.25e3·113-s − 1.90e4·117-s + 5.55e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 2.33e4·153-s + 157-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 2.08·13-s + 1.49·17-s + 2/5·25-s − 2.56·29-s − 0.783·37-s + 1.29·41-s + 1.94·49-s + 1.67·53-s − 0.268·61-s − 2.48·73-s + 1/3·81-s − 4.36·89-s + 1.30·97-s + 3.81·101-s − 0.388·109-s − 0.646·113-s − 1.38·117-s + 3.79·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s − 0.996·153-s + 4.05e−5·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.9281140474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9281140474\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 4660 T^{2} + 11126502 T^{4} - 4660 p^{8} T^{6} + p^{16} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 55540 T^{2} + 1199818662 T^{4} - 55540 p^{8} T^{6} + p^{16} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 176 T + 60366 T^{2} - 176 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 216 T + 138206 T^{2} - 216 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 501508 T^{2} + 96751382598 T^{4} - 501508 p^{8} T^{6} + p^{16} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 809764 T^{2} + 297972204486 T^{4} - 809764 p^{8} T^{6} + p^{16} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 1080 T + 1485662 T^{2} + 1080 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2663620 T^{2} + 3467278382982 T^{4} - 2663620 p^{8} T^{6} + p^{16} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 536 T + 2195646 T^{2} + 536 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 1092 T + 3357638 T^{2} - 1092 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 5563108 T^{2} + 31112664458118 T^{4} - 5563108 p^{8} T^{6} + p^{16} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 15975460 T^{2} + 111165388324422 T^{4} - 15975460 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 2352 T + 17123438 T^{2} - 2352 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 11029300 T^{2} + 217832022863142 T^{4} - 11029300 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 500 T + 348462 p T^{2} + 500 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 35762788 T^{2} + 1089238261984518 T^{4} - 35762788 p^{8} T^{6} + p^{16} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 101457220 T^{2} + 3864896804975622 T^{4} - 101457220 p^{8} T^{6} + p^{16} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 6628 T + 17217078 T^{2} + 6628 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 71404228 T^{2} + 2862851988650118 T^{4} - 71404228 p^{8} T^{6} + p^{16} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 139920868 T^{2} + 9351476705394438 T^{4} - 139920868 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 17268 T + 160268438 T^{2} + 17268 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 6116 T + 185257926 T^{2} - 6116 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
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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
−8.379734624223922302979410646827, −7.68177557562366417374001672077, −7.64595034104109356509870685398, −7.41906020116051346567954943813, −7.15078018325752905818097174875, −7.00882018310075443652939206179, −6.28427847169667462846546164995, −6.25036560296810863864725715746, −5.88374659197735065730183360219, −5.75290040238705269888941298682, −5.42232156551586994704577225193, −5.40301268041789368802340569216, −4.80235343013929124687622948586, −4.38274689188553899480271634906, −4.08258485747223673580515423027, −3.79535298906650987543811863162, −3.46412002423267765518229726054, −3.32016736067331089059625527910, −2.88705619384144531490569131582, −2.31107590740901471881317150893, −2.14427116370555646381082030759, −1.46604744342701873563905125643, −1.07619852082101823811904620779, −0.984083585406517058342246148721, −0.14385707338379969083887142914,
0.14385707338379969083887142914, 0.984083585406517058342246148721, 1.07619852082101823811904620779, 1.46604744342701873563905125643, 2.14427116370555646381082030759, 2.31107590740901471881317150893, 2.88705619384144531490569131582, 3.32016736067331089059625527910, 3.46412002423267765518229726054, 3.79535298906650987543811863162, 4.08258485747223673580515423027, 4.38274689188553899480271634906, 4.80235343013929124687622948586, 5.40301268041789368802340569216, 5.42232156551586994704577225193, 5.75290040238705269888941298682, 5.88374659197735065730183360219, 6.25036560296810863864725715746, 6.28427847169667462846546164995, 7.00882018310075443652939206179, 7.15078018325752905818097174875, 7.41906020116051346567954943813, 7.64595034104109356509870685398, 7.68177557562366417374001672077, 8.379734624223922302979410646827
