| L(s) = 1 | − 16-s + 40·25-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 104·169-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 + 239-s + 241-s + 251-s + ⋯ |
| L(s) = 1 | − 1/4·16-s + 8·25-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 − 8·169-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 + 0.0646·239-s + 0.0644·241-s + 0.0631·251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.409517167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.409517167\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T^{4} + p^{4} T^{8} \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 - p T^{2} )^{8} \) |
| 11 | \( ( 1 - 206 T^{4} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + p T^{2} )^{8} \) |
| 17 | \( ( 1 - p T^{2} )^{8} \) |
| 19 | \( ( 1 - p T^{2} )^{8} \) |
| 23 | \( ( 1 - 734 T^{4} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 1234 T^{4} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - p T^{2} )^{8} \) |
| 37 | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - p T^{2} )^{8} \) |
| 43 | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + p T^{2} )^{8} \) |
| 53 | \( ( 1 - 5582 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + p T^{2} )^{8} \) |
| 61 | \( ( 1 + p T^{2} )^{8} \) |
| 67 | \( ( 1 - 4 T + p T^{2} )^{4}( 1 + 4 T + p T^{2} )^{4} \) |
| 71 | \( ( 1 + 2914 T^{4} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + p T^{2} )^{8} \) |
| 79 | \( ( 1 - 8 T + p T^{2} )^{4}( 1 + 8 T + p T^{2} )^{4} \) |
| 83 | \( ( 1 + p T^{2} )^{8} \) |
| 89 | \( ( 1 - p T^{2} )^{8} \) |
| 97 | \( ( 1 + p T^{2} )^{8} \) |
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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
−3.91994937968984945359937857199, −3.76811134878409403669927393250, −3.65144859355296760302242581104, −3.51956736401028740466167290689, −3.40918517280381078706709738677, −3.20573116126984107624550954939, −3.11399202297402287835073104776, −2.92614850017699024705418321777, −2.91191697271082113680917885927, −2.90850486927284505701811177868, −2.63173646203382058077457704177, −2.53111625400919786979314569549, −2.48131658203164216809990923619, −2.24579351384033417156779402665, −2.18229772609466143605242245771, −1.94264850714508135166459931508, −1.63726291331363011581626885102, −1.48428093314435314218597358849, −1.45328027664496390013844445215, −1.20723827344751300427739586131, −0.953459356125083143976812203477, −0.914093875373577056563662573168, −0.867907152865859272279176720612, −0.54024605251455427120994492065, −0.11909148935263438302611734128,
0.11909148935263438302611734128, 0.54024605251455427120994492065, 0.867907152865859272279176720612, 0.914093875373577056563662573168, 0.953459356125083143976812203477, 1.20723827344751300427739586131, 1.45328027664496390013844445215, 1.48428093314435314218597358849, 1.63726291331363011581626885102, 1.94264850714508135166459931508, 2.18229772609466143605242245771, 2.24579351384033417156779402665, 2.48131658203164216809990923619, 2.53111625400919786979314569549, 2.63173646203382058077457704177, 2.90850486927284505701811177868, 2.91191697271082113680917885927, 2.92614850017699024705418321777, 3.11399202297402287835073104776, 3.20573116126984107624550954939, 3.40918517280381078706709738677, 3.51956736401028740466167290689, 3.65144859355296760302242581104, 3.76811134878409403669927393250, 3.91994937968984945359937857199
Plot not available for L-functions of degree greater than 10.
