| L(s) = 1 | + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + 241-s + 251-s + 257-s + 263-s + 269-s + ⋯ |
| L(s) = 1 | + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + 241-s + 251-s + 257-s + 263-s + 269-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2821188653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2821188653\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T^{16} \) |
| 17 | \( 1 + T^{16} \) |
| good | 2 | \( ( 1 + T^{16} )^{2} \) |
| 7 | \( ( 1 + T^{16} )^{2} \) |
| 11 | \( ( 1 + T^{16} )^{2} \) |
| 13 | \( ( 1 + T^{4} )^{8} \) |
| 19 | \( ( 1 + T^{2} )^{8}( 1 + T^{4} )^{4} \) |
| 23 | \( ( 1 + T^{16} )^{2} \) |
| 29 | \( ( 1 + T^{16} )^{2} \) |
| 31 | \( ( 1 + T^{4} )^{4}( 1 + T^{8} )^{2} \) |
| 37 | \( ( 1 + T^{16} )^{2} \) |
| 41 | \( ( 1 + T^{16} )^{2} \) |
| 43 | \( ( 1 + T^{8} )^{4} \) |
| 47 | \( ( 1 + T^{16} )^{2} \) |
| 53 | \( ( 1 + T^{16} )^{2} \) |
| 59 | \( ( 1 + T^{8} )^{4} \) |
| 61 | \( ( 1 + T^{2} )^{8}( 1 + T^{8} )^{2} \) |
| 67 | \( ( 1 + T^{2} )^{16} \) |
| 71 | \( ( 1 + T^{16} )^{2} \) |
| 73 | \( ( 1 + T^{16} )^{2} \) |
| 79 | \( ( 1 + T^{4} )^{4}( 1 + T^{8} )^{2} \) |
| 83 | \( ( 1 + T^{16} )^{2} \) |
| 89 | \( ( 1 + T^{4} )^{8} \) |
| 97 | \( ( 1 + T^{16} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.97730045998207573032930133339, −2.94710363602184265221011741971, −2.84930623426575695339333047017, −2.82012841196487127354143360108, −2.81836618054093303435389349861, −2.72097161958540813175416436488, −2.68605633806220370917440214395, −2.38917756139351211811678578121, −2.28497297713887498148155707920, −2.28369476668750028460570189517, −2.18680474008867270906849577472, −2.12610440049152456759970863715, −2.05337541996567255759698809051, −2.01592566589998546916894904115, −1.79538733170427666930922500126, −1.70468501177738606497302629244, −1.64524018631882813122119543710, −1.61598307854116221324870686483, −1.55814090834279787235449273614, −1.29638118279247382254072612825, −1.25131028205294972961245863721, −1.06554931058569505073489949014, −0.862381926309192441169465725467, −0.829679063317857432635731126611, −0.816155180189329016130218229162,
0.816155180189329016130218229162, 0.829679063317857432635731126611, 0.862381926309192441169465725467, 1.06554931058569505073489949014, 1.25131028205294972961245863721, 1.29638118279247382254072612825, 1.55814090834279787235449273614, 1.61598307854116221324870686483, 1.64524018631882813122119543710, 1.70468501177738606497302629244, 1.79538733170427666930922500126, 2.01592566589998546916894904115, 2.05337541996567255759698809051, 2.12610440049152456759970863715, 2.18680474008867270906849577472, 2.28369476668750028460570189517, 2.28497297713887498148155707920, 2.38917756139351211811678578121, 2.68605633806220370917440214395, 2.72097161958540813175416436488, 2.81836618054093303435389349861, 2.82012841196487127354143360108, 2.84930623426575695339333047017, 2.94710363602184265221011741971, 2.97730045998207573032930133339
Plot not available for L-functions of degree greater than 10.
