| L(s) = 1 | + 4·5-s − 28·13-s − 64·17-s − 48·25-s − 32·29-s + 32·37-s + 80·41-s − 14·49-s + 184·53-s + 204·61-s − 112·65-s + 184·73-s − 256·85-s + 56·89-s + 32·97-s + 20·101-s − 304·109-s + 256·113-s + 404·121-s − 228·125-s + 127-s + 131-s + 137-s + 139-s − 128·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 4/5·5-s − 2.15·13-s − 3.76·17-s − 1.91·25-s − 1.10·29-s + 0.864·37-s + 1.95·41-s − 2/7·49-s + 3.47·53-s + 3.34·61-s − 1.72·65-s + 2.52·73-s − 3.01·85-s + 0.629·89-s + 0.329·97-s + 0.198·101-s − 2.78·109-s + 2.26·113-s + 3.33·121-s − 1.82·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 0.882·145-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.169057413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.169057413\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 5 | $D_{4}$ | \( ( 1 - 2 T + 6 p T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 404 T^{2} + 68742 T^{4} - 404 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 14 T + 198 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 32 T + 750 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 464 T^{2} + 112782 T^{4} - 464 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1508 T^{2} + 1042182 T^{4} - 1508 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 16 T - 354 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 716 T^{2} - 69018 T^{4} + 716 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 16 T + 2718 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 40 T + 3006 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 1940 T^{2} + 7476102 T^{4} - 1940 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 4084 T^{2} + 12949350 T^{4} - 4084 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 + 2032 T^{2} + 19804878 T^{4} + 2032 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 102 T + 9518 T^{2} - 102 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 572 T^{2} + 33416742 T^{4} + 572 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 13700 T^{2} + 92907462 T^{4} - 13700 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 92 T + 4374 T^{2} - 92 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 22084 T^{2} + 198084102 T^{4} - 22084 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 4016 T^{2} + 20075982 T^{4} - 4016 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 28 T + 10662 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 18798 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.96437929388530816343338105365, −6.67162215498538434350635486618, −6.59658472117011207014284254215, −6.28235600341418323792557171600, −5.98958999562961626657712609761, −5.74910426781801161885803250506, −5.69811920547307987213930881695, −5.18754066879053178935875834732, −5.12134368524641425203300949212, −5.03295275712786438684382678616, −4.40538040930544526783209539512, −4.35905451597666429419545074423, −4.26933794483613608464010588915, −3.91462940137091967379236282324, −3.61336759722597253178392897677, −3.49982273972534330408179821667, −2.65776526896979008214188103758, −2.48448520404120224032988436648, −2.48214970180704707524857604304, −2.18365915235690770388813521916, −1.97100462086913743415252116760, −1.81347984492827358904286994932, −0.894357623475546984941788996905, −0.65063300616288568048817073117, −0.19605667084860160138295023083,
0.19605667084860160138295023083, 0.65063300616288568048817073117, 0.894357623475546984941788996905, 1.81347984492827358904286994932, 1.97100462086913743415252116760, 2.18365915235690770388813521916, 2.48214970180704707524857604304, 2.48448520404120224032988436648, 2.65776526896979008214188103758, 3.49982273972534330408179821667, 3.61336759722597253178392897677, 3.91462940137091967379236282324, 4.26933794483613608464010588915, 4.35905451597666429419545074423, 4.40538040930544526783209539512, 5.03295275712786438684382678616, 5.12134368524641425203300949212, 5.18754066879053178935875834732, 5.69811920547307987213930881695, 5.74910426781801161885803250506, 5.98958999562961626657712609761, 6.28235600341418323792557171600, 6.59658472117011207014284254215, 6.67162215498538434350635486618, 6.96437929388530816343338105365
