| L(s) = 1 | − 2·4-s − 5·9-s + 3·16-s + 10·36-s − 48·41-s − 4·49-s + 24·59-s − 12·64-s − 4·79-s + 9·81-s − 72·89-s + 48·101-s + 68·109-s + 74·121-s + 127-s + 131-s + 137-s + 139-s − 15·144-s + 149-s + 151-s + 157-s + 163-s + 96·164-s + 167-s − 2·169-s + 173-s + ⋯ |
| L(s) = 1 | − 4-s − 5/3·9-s + 3/4·16-s + 5/3·36-s − 7.49·41-s − 4/7·49-s + 3.12·59-s − 3/2·64-s − 0.450·79-s + 81-s − 7.63·89-s + 4.77·101-s + 6.51·109-s + 6.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5/4·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 7.49·164-s + 0.0773·167-s − 0.153·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.017937196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.017937196\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 2 | \( ( 1 + T^{2} + p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 37 T^{2} + 576 T^{4} - 37 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + T^{2} + 264 T^{4} + p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 47 T^{2} + 1056 T^{4} - 47 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 8 T + p T^{2} )^{4}( 1 + 8 T + p T^{2} )^{4} \) |
| 23 | \( ( 1 + 16 T^{2} - 66 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 97 T^{2} + 3960 T^{4} - 97 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 80 T^{2} + 4206 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 6 T + p T^{2} )^{8} \) |
| 43 | \( ( 1 - 104 T^{2} + 6270 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 131 T^{2} + 8040 T^{4} - 131 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 136 T^{2} + 9054 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 14 T + p T^{2} )^{4}( 1 + 14 T + p T^{2} )^{4} \) |
| 67 | \( ( 1 - 200 T^{2} + 18846 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 100 T^{2} + 4134 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + T + 150 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 4 T^{2} - 5226 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 18 T + 226 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 265 T^{2} + 32736 T^{4} + 265 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−4.78385864901493550936498246268, −4.76895803325990689116006038205, −4.63080033300410024921108059716, −4.28091842290024955264614465966, −4.21288468558257730661323569360, −4.12031015900314002493891349209, −3.81768002288004657710250210751, −3.72306542630018199758655764664, −3.56734604533631380935670044955, −3.44505402013636177258084803891, −3.35021021375821708529831650545, −3.08155152125132839770837710761, −3.02544415062221868839336843454, −3.02407665648199169150376965414, −2.78236448071119062282743707708, −2.46843248416150086766864586030, −2.24759570326835229254030108210, −1.99685524035399596224557046793, −1.91826275442834202383096499302, −1.75514808626944007314771185280, −1.48074031906058615434276580132, −1.36742944608551150808558186017, −0.838721301636811534203425048740, −0.39382171974119488230983833991, −0.34852288831556581176598045684,
0.34852288831556581176598045684, 0.39382171974119488230983833991, 0.838721301636811534203425048740, 1.36742944608551150808558186017, 1.48074031906058615434276580132, 1.75514808626944007314771185280, 1.91826275442834202383096499302, 1.99685524035399596224557046793, 2.24759570326835229254030108210, 2.46843248416150086766864586030, 2.78236448071119062282743707708, 3.02407665648199169150376965414, 3.02544415062221868839336843454, 3.08155152125132839770837710761, 3.35021021375821708529831650545, 3.44505402013636177258084803891, 3.56734604533631380935670044955, 3.72306542630018199758655764664, 3.81768002288004657710250210751, 4.12031015900314002493891349209, 4.21288468558257730661323569360, 4.28091842290024955264614465966, 4.63080033300410024921108059716, 4.76895803325990689116006038205, 4.78385864901493550936498246268
Plot not available for L-functions of degree greater than 10.
