| L(s) = 1 | + 14·9-s + 100·25-s − 196·49-s − 568·73-s + 115·81-s − 376·97-s + 92·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 676·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 1.40e3·225-s + ⋯ |
| L(s) = 1 | + 14/9·9-s + 4·25-s − 4·49-s − 7.78·73-s + 1.41·81-s − 3.87·97-s + 0.760·121-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 − 4·169-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 + 56/9·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.09556955500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09556955500\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 14 T^{2} + p^{4} T^{4} \) |
| good | 5 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2}( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 434 T^{2} + p^{4} T^{4} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )^{2}( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 3502 T^{2} + p^{4} T^{4} )^{2} \) |
| 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 | $C_2^2$ | \( ( 1 - 238 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 5134 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 142 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 11186 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 146 T + p^{2} T^{2} )^{2}( 1 + 146 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 94 T + p^{2} T^{2} )^{4} \) |
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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
−7.00879917454896232851109247840, −6.94029189952588883552514372375, −6.80992643578864394722428814452, −6.76040856783115788878324961051, −6.23181953872864823937099783579, −6.15659636866224041290773310214, −5.83815802319003284027986191977, −5.45073797576647302856782005838, −5.38126678357049588843816289882, −4.93195712800416955711357331986, −4.58995259995305007823911337123, −4.58491949272348302023670203975, −4.49799617493307901531764614797, −4.24717716778894650797233325681, −3.64266858952533784618349501352, −3.35805764807088626954579788717, −3.28786986920726198447903194643, −2.80728873216392143731892900783, −2.70411194709296404025987565066, −2.39917421385043750282690161181, −1.54805795712717115100405820466, −1.48952268650664630491492644701, −1.36875101064176717597572361048, −0.950199002863151206450412548067, −0.04569525081837175703058432778,
0.04569525081837175703058432778, 0.950199002863151206450412548067, 1.36875101064176717597572361048, 1.48952268650664630491492644701, 1.54805795712717115100405820466, 2.39917421385043750282690161181, 2.70411194709296404025987565066, 2.80728873216392143731892900783, 3.28786986920726198447903194643, 3.35805764807088626954579788717, 3.64266858952533784618349501352, 4.24717716778894650797233325681, 4.49799617493307901531764614797, 4.58491949272348302023670203975, 4.58995259995305007823911337123, 4.93195712800416955711357331986, 5.38126678357049588843816289882, 5.45073797576647302856782005838, 5.83815802319003284027986191977, 6.15659636866224041290773310214, 6.23181953872864823937099783579, 6.76040856783115788878324961051, 6.80992643578864394722428814452, 6.94029189952588883552514372375, 7.00879917454896232851109247840
