| L(s) = 1 | + 4·2-s + 11·4-s + 8·5-s − 10·7-s + 22·8-s + 6·9-s + 32·10-s − 8·11-s − 8·13-s − 40·14-s + 36·16-s + 4·17-s + 24·18-s + 10·19-s + 88·20-s − 32·22-s + 10·23-s + 38·25-s − 32·26-s − 110·28-s + 24·29-s − 18·31-s + 52·32-s + 16·34-s − 80·35-s + 66·36-s − 6·37-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 11/2·4-s + 3.57·5-s − 3.77·7-s + 7.77·8-s + 2·9-s + 10.1·10-s − 2.41·11-s − 2.21·13-s − 10.6·14-s + 9·16-s + 0.970·17-s + 5.65·18-s + 2.29·19-s + 19.6·20-s − 6.82·22-s + 2.08·23-s + 38/5·25-s − 6.27·26-s − 20.7·28-s + 4.45·29-s − 3.23·31-s + 9.19·32-s + 2.74·34-s − 13.5·35-s + 11·36-s − 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(18.39401918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.39401918\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p^{2} T + 5 T^{2} + p T^{3} - 11 T^{4} + p^{2} T^{5} + 5 p^{2} T^{6} - p^{5} T^{7} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 8 T + 20 T^{2} - 36 T^{3} - 361 T^{4} - 36 p T^{5} + 20 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^3$ | \( 1 - 4 T + 8 T^{2} + 104 T^{3} - 497 T^{4} + 104 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 10 T + 40 T^{2} - 220 T^{3} + 1339 T^{4} - 220 p T^{5} + 40 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 10 T + 50 T^{2} - 340 T^{3} + 2191 T^{4} - 340 p T^{5} + 50 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 18 T + 172 T^{2} + 1152 T^{3} + 6483 T^{4} + 1152 p T^{5} + 172 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 + 6 T + 18 T^{2} - 336 T^{3} - 2377 T^{4} - 336 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 12 T + 106 T^{2} + 696 T^{3} + 3651 T^{4} + 696 p T^{5} + 106 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 6 T + 45 T^{2} - 366 T^{3} + 1328 T^{4} - 366 p T^{5} + 45 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 24 T + 369 T^{2} - 3792 T^{3} + 30092 T^{4} - 3792 p T^{5} + 369 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 10 T + 74 T^{2} + 520 T^{3} + 2551 T^{4} + 520 p T^{5} + 74 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 106 T^{2} + 7755 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 36 T + 661 T^{2} + 8244 T^{3} + 75072 T^{4} + 8244 p T^{5} + 661 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 16 T + 65 T^{2} - 960 T^{3} - 14284 T^{4} - 960 p T^{5} + 65 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 + 18 T + 90 T^{2} - 840 T^{3} - 14929 T^{4} - 840 p T^{5} + 90 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 12 T + 154 T^{2} + 1272 T^{3} + 8787 T^{4} + 1272 p T^{5} + 154 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^3$ | \( 1 + 2 T + 2 T^{2} - 328 T^{3} - 7217 T^{4} - 328 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 8 T - 103 T^{2} + 88 T^{3} + 14272 T^{4} + 88 p T^{5} - 103 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 14 T + 50 T^{2} - 1500 T^{3} - 23089 T^{4} - 1500 p T^{5} + 50 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.962166399069444923324253075216, −7.75112521767558151734428877111, −7.47604612118324419605785404941, −7.41586425351249769546076782423, −7.30778070142123944497219917324, −6.84424215203775369345220950607, −6.70259610839748692798263668379, −6.64661266107380402057013430666, −6.42782946492468748533848264581, −5.81640516647960740647113602964, −5.74192982056113064527463401039, −5.63842017804906777853234528946, −5.36587030356323068513949997206, −5.30347686404306712511408696729, −4.69379759659259130746954100646, −4.49095717154789528093087207369, −4.45915260506102175926438040120, −3.30990931373935617882119816718, −3.17246637769032109031987996950, −3.00536102888674295639694855635, −2.84903739902525194946643129076, −2.64791932917889440177018384654, −2.48634501539255169392026058589, −1.63071287078507208076661067811, −1.30546397913012155492974109565,
1.30546397913012155492974109565, 1.63071287078507208076661067811, 2.48634501539255169392026058589, 2.64791932917889440177018384654, 2.84903739902525194946643129076, 3.00536102888674295639694855635, 3.17246637769032109031987996950, 3.30990931373935617882119816718, 4.45915260506102175926438040120, 4.49095717154789528093087207369, 4.69379759659259130746954100646, 5.30347686404306712511408696729, 5.36587030356323068513949997206, 5.63842017804906777853234528946, 5.74192982056113064527463401039, 5.81640516647960740647113602964, 6.42782946492468748533848264581, 6.64661266107380402057013430666, 6.70259610839748692798263668379, 6.84424215203775369345220950607, 7.30778070142123944497219917324, 7.41586425351249769546076782423, 7.47604612118324419605785404941, 7.75112521767558151734428877111, 8.962166399069444923324253075216
