| L(s) = 1 | − 32·25-s + 64·43-s + 96·67-s − 32·79-s + 72·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-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 + ⋯ |
| L(s) = 1 | − 6.39·25-s + 9.75·43-s + 11.7·67-s − 3.60·79-s + 6.54·121-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 + 2.46·169-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 + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(17.62472166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.62472166\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 + 16 T^{2} + 112 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 16 T^{2} + 240 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 48 T^{2} + 1056 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 60 T^{2} + 1590 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 68 T^{2} + 2086 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 92 T^{2} + 3670 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 108 T^{2} + 4806 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 72 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 160 T^{2} + 9760 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 16 T + 142 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 108 T^{2} + 5766 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 144 T^{2} + 9650 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 28 T^{2} - 2090 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 192 T^{2} + 16080 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 24 T + 270 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 20 T^{2} - 2618 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 80 T^{2} + 160 p T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 12 T^{2} + 13302 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 96 T^{2} + 192 p T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 48 T^{2} + 17472 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.25212492336481653042725442486, −3.22314560473354520422855492147, −3.05498495756770576693967100254, −2.77102302475834780581642357033, −2.58868076988852265714259441073, −2.51325567896974488120185247839, −2.49195359641122168953512917577, −2.48041029602021954591707848378, −2.46550791855269039127835630175, −2.14464266967215017020365013460, −2.11982371324113559413473222474, −2.00231777705159830585687820484, −1.90581304942271446635105407005, −1.85250720486151772156386418781, −1.80710000872829194208964011708, −1.67101656421554150126651717902, −1.25994346588104793739218243443, −1.09889722979500628101832015016, −0.929698539399304933380671559837, −0.909189671312924400481633080935, −0.852815458180415332111289102833, −0.59413863070049469584656840832, −0.58715726335337155423607950905, −0.31997222049992232466695830649, −0.25855204555991296481152546901,
0.25855204555991296481152546901, 0.31997222049992232466695830649, 0.58715726335337155423607950905, 0.59413863070049469584656840832, 0.852815458180415332111289102833, 0.909189671312924400481633080935, 0.929698539399304933380671559837, 1.09889722979500628101832015016, 1.25994346588104793739218243443, 1.67101656421554150126651717902, 1.80710000872829194208964011708, 1.85250720486151772156386418781, 1.90581304942271446635105407005, 2.00231777705159830585687820484, 2.11982371324113559413473222474, 2.14464266967215017020365013460, 2.46550791855269039127835630175, 2.48041029602021954591707848378, 2.49195359641122168953512917577, 2.51325567896974488120185247839, 2.58868076988852265714259441073, 2.77102302475834780581642357033, 3.05498495756770576693967100254, 3.22314560473354520422855492147, 3.25212492336481653042725442486
Plot not available for L-functions of degree greater than 10.
