| L(s) = 1 | + 16·23-s − 8·25-s − 48·47-s + 32·49-s − 16·71-s − 16·73-s + 24·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | + 3.33·23-s − 8/5·25-s − 7.00·47-s + 32/7·49-s − 1.89·71-s − 1.87·73-s + 2.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9743474062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9743474062\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 4 T^{2} + 22 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 16 T^{2} + 130 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 12 T^{2} + 150 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 20 T^{2} + 310 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + 12 T^{2} + 246 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 36 T^{2} + 1974 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 80 T^{2} + 3234 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 84 T^{2} + 3990 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2}( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | \( ( 1 + 44 T^{2} + 2134 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + 132 T^{2} + 9942 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 108 T^{2} + 7830 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 180 T^{2} + 15030 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 52 T^{2} + 9142 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 16 T^{2} - 7454 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 172 T^{2} + 21046 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 128 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 66 T^{2} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.38707805422878720317940105659, −3.17543801313230146836065250444, −3.13159513495891163170917341557, −3.12763303822054493917142841911, −2.99823846052949708356775342081, −2.87697194798294363656192014435, −2.85501693715538848303868001905, −2.60046963557257122331755587858, −2.41706462211640068492784210274, −2.36044586121742563949562274375, −2.28392367484450526780399521688, −2.26221791983709642465466451753, −2.01541471711805719994273487735, −1.73602605631198394780791844237, −1.59849970750628766487469758507, −1.58763722452018298121780660796, −1.54428046636117024254612366167, −1.41993266015992950469561848269, −1.25317224035400644878349259476, −1.00229815699266280318978667219, −0.908110282712652384243857631432, −0.66861238225639759141956693853, −0.44449161687371357849270942762, −0.40605534050304296924723478286, −0.06916462486139464884913268747,
0.06916462486139464884913268747, 0.40605534050304296924723478286, 0.44449161687371357849270942762, 0.66861238225639759141956693853, 0.908110282712652384243857631432, 1.00229815699266280318978667219, 1.25317224035400644878349259476, 1.41993266015992950469561848269, 1.54428046636117024254612366167, 1.58763722452018298121780660796, 1.59849970750628766487469758507, 1.73602605631198394780791844237, 2.01541471711805719994273487735, 2.26221791983709642465466451753, 2.28392367484450526780399521688, 2.36044586121742563949562274375, 2.41706462211640068492784210274, 2.60046963557257122331755587858, 2.85501693715538848303868001905, 2.87697194798294363656192014435, 2.99823846052949708356775342081, 3.12763303822054493917142841911, 3.13159513495891163170917341557, 3.17543801313230146836065250444, 3.38707805422878720317940105659
Plot not available for L-functions of degree greater than 10.
