| L(s) = 1 | + 100·25-s − 188·49-s − 184·73-s − 8·97-s − 484·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 292·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 4·25-s − 3.83·49-s − 2.52·73-s − 0.0824·97-s − 4·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.72·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.06426642778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06426642778\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2}( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + 146 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )^{2}( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2062 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 3214 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 1966 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 5906 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )^{2}( 1 + 142 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{4} \) |
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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
−6.23248525516154022199607206244, −6.14353094764651005355899274442, −5.90605606682201588505037692799, −5.32866737403374284677584309886, −5.31212491811102545489852717806, −5.26077982744396780234714156184, −4.86973399648162075734635453368, −4.83247014835427112347868630950, −4.46584739313504489436522546372, −4.42856291239911575569499403689, −4.20743777020550089659871819793, −3.69844199999444156628778317366, −3.58586603657582804014793119464, −3.28834548187421380994412350410, −3.16570136011276093075230831840, −2.85985099796296523317651667628, −2.72572708795792804742666724951, −2.40006293295397425741309633681, −2.22939913359474318135192884964, −1.68045250254813599180321532949, −1.40893859266502846699019968045, −1.18065369530714239652909557510, −1.14867004151208707733286682957, −0.54124297949325164943205189098, −0.03188909107629850205720401132,
0.03188909107629850205720401132, 0.54124297949325164943205189098, 1.14867004151208707733286682957, 1.18065369530714239652909557510, 1.40893859266502846699019968045, 1.68045250254813599180321532949, 2.22939913359474318135192884964, 2.40006293295397425741309633681, 2.72572708795792804742666724951, 2.85985099796296523317651667628, 3.16570136011276093075230831840, 3.28834548187421380994412350410, 3.58586603657582804014793119464, 3.69844199999444156628778317366, 4.20743777020550089659871819793, 4.42856291239911575569499403689, 4.46584739313504489436522546372, 4.83247014835427112347868630950, 4.86973399648162075734635453368, 5.26077982744396780234714156184, 5.31212491811102545489852717806, 5.32866737403374284677584309886, 5.90605606682201588505037692799, 6.14353094764651005355899274442, 6.23248525516154022199607206244
