| L(s) = 1 | + 2·2-s + 3·4-s + 8·8-s − 4·9-s + 9·16-s − 8·18-s − 4·25-s − 16·29-s + 6·32-s − 12·36-s + 16·37-s + 18·49-s − 8·50-s + 16·53-s − 32·58-s + 11·64-s − 32·72-s + 32·74-s + 8·81-s + 36·98-s − 12·100-s + 32·106-s − 24·109-s + 16·113-s − 48·116-s + 32·121-s + 127-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 2.82·8-s − 4/3·9-s + 9/4·16-s − 1.88·18-s − 4/5·25-s − 2.97·29-s + 1.06·32-s − 2·36-s + 2.63·37-s + 18/7·49-s − 1.13·50-s + 2.19·53-s − 4.20·58-s + 11/8·64-s − 3.77·72-s + 3.71·74-s + 8/9·81-s + 3.63·98-s − 6/5·100-s + 3.10·106-s − 2.29·109-s + 1.50·113-s − 4.45·116-s + 2.90·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.209658134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.209658134\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T - p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | \( ( 1 + T^{2} )^{4} \) |
| 7 | \( 1 - 18 T^{2} + 162 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 3 | \( ( 1 + 2 T^{2} + 2 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 16 T^{2} + 238 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 16 T^{2} + 30 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 48 T^{2} + 1230 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 18 T^{2} + 1122 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 31 | \( ( 1 + 68 T^{2} + 2806 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 41 | \( ( 1 - 112 T^{2} + 5886 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 114 T^{2} + 6114 T^{4} - 114 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 62 T^{2} + 4002 T^{4} + 62 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 59 | \( ( 1 + 112 T^{2} + 6766 T^{4} + 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 50 T^{2} + 610 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 192 T^{2} + 19230 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 32 T^{2} + 5406 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 296 T^{2} + 34318 T^{4} - 296 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 322 T^{2} + 39682 T^{4} + 322 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 336 T^{2} + 46430 T^{4} - 336 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−6.05219150961933518203239323790, −5.92463275837571911969807782698, −5.61147730785555401941343622108, −5.54514950020594540435832478849, −5.51948561684313595599906721845, −5.23008366464435859465527503028, −5.18409885318345428302534497699, −5.13302341777082330521003232403, −4.78406598656606920428512972504, −4.44548117566682680191761342058, −4.37313079396885604715450589352, −4.14417934128257587207001844612, −3.93235680772044932606481856573, −3.93193478839480476454931162721, −3.92712216470713048158606206356, −3.49272063160832446146007279785, −3.34465993432659366823520021504, −2.85398824592260804443183247156, −2.72501349137424717206921899335, −2.52697532430818881695725016538, −2.46967958064662831402488955822, −1.96614805819024225451468983340, −1.88383855470931835637114215108, −1.54326506653660066469279922125, −0.851456247322789984511265991302,
0.851456247322789984511265991302, 1.54326506653660066469279922125, 1.88383855470931835637114215108, 1.96614805819024225451468983340, 2.46967958064662831402488955822, 2.52697532430818881695725016538, 2.72501349137424717206921899335, 2.85398824592260804443183247156, 3.34465993432659366823520021504, 3.49272063160832446146007279785, 3.92712216470713048158606206356, 3.93193478839480476454931162721, 3.93235680772044932606481856573, 4.14417934128257587207001844612, 4.37313079396885604715450589352, 4.44548117566682680191761342058, 4.78406598656606920428512972504, 5.13302341777082330521003232403, 5.18409885318345428302534497699, 5.23008366464435859465527503028, 5.51948561684313595599906721845, 5.54514950020594540435832478849, 5.61147730785555401941343622108, 5.92463275837571911969807782698, 6.05219150961933518203239323790
Plot not available for L-functions of degree greater than 10.
