| L(s) = 1 | − 64·25-s − 360·49-s + 320·73-s − 320·97-s + 120·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.08e3·169-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 | − 2.55·25-s − 7.34·49-s + 4.38·73-s − 3.29·97-s + 0.991·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 6.39·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6828679312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6828679312\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 16 T^{2} + p^{4} T^{4} )^{4} \) |
| 7 | \( ( 1 + 90 T^{2} + p^{4} T^{4} )^{4} \) |
| 11 | \( ( 1 - 30 T^{2} + p^{4} T^{4} )^{4} \) |
| 13 | \( ( 1 - 270 T^{2} + p^{4} T^{4} )^{4} \) |
| 17 | \( ( 1 - 528 T^{2} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 - 30 T + p^{2} T^{2} )^{4}( 1 + 30 T + p^{2} T^{2} )^{4} \) |
| 23 | \( ( 1 - 658 T^{2} + p^{4} T^{4} )^{4} \) |
| 29 | \( ( 1 + 832 T^{2} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 + 122 T^{2} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 - 290 T^{2} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 - 2640 T^{2} + p^{4} T^{4} )^{4} \) |
| 43 | \( ( 1 - p T )^{8}( 1 + p T )^{8} \) |
| 47 | \( ( 1 - 2482 T^{2} + p^{4} T^{4} )^{4} \) |
| 53 | \( ( 1 + 4768 T^{2} + p^{4} T^{4} )^{4} \) |
| 59 | \( ( 1 + 2610 T^{2} + p^{4} T^{4} )^{4} \) |
| 61 | \( ( 1 - 642 T^{2} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 - 66 T + p^{2} T^{2} )^{4}( 1 + 66 T + p^{2} T^{2} )^{4} \) |
| 71 | \( ( 1 - 82 T^{2} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 - 40 T + p^{2} T^{2} )^{8} \) |
| 79 | \( ( 1 - 3718 T^{2} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 + 6978 T^{2} + p^{4} T^{4} )^{4} \) |
| 89 | \( ( 1 - 2720 T^{2} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 + 40 T + p^{2} T^{2} )^{8} \) |
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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
−3.91624214104102468778459677922, −3.82507306757703508597721357712, −3.60378502237672287487362315844, −3.57754217924805269470057020107, −3.41046824069495776981128761145, −3.31158705368128230376266593473, −3.28691638143702816522427601846, −3.13002595626263879655497781263, −2.89617252801749625577047888618, −2.65229207769409095821805644027, −2.64699676082283425349733777957, −2.53757433578970764763522686177, −2.18892979704175407990317290704, −2.03680618682403612314357916338, −2.00946055426531084801465804490, −1.96432654080240081258775579322, −1.59085298506657218532344577161, −1.56790354967416291110838031234, −1.38513249592748879950850489441, −1.28450533022463371865202654589, −0.924892892703000040856471765640, −0.77506951424417257569674806744, −0.49471511887167158344946774506, −0.16562021406161888191630680934, −0.14159631471071321700838570546,
0.14159631471071321700838570546, 0.16562021406161888191630680934, 0.49471511887167158344946774506, 0.77506951424417257569674806744, 0.924892892703000040856471765640, 1.28450533022463371865202654589, 1.38513249592748879950850489441, 1.56790354967416291110838031234, 1.59085298506657218532344577161, 1.96432654080240081258775579322, 2.00946055426531084801465804490, 2.03680618682403612314357916338, 2.18892979704175407990317290704, 2.53757433578970764763522686177, 2.64699676082283425349733777957, 2.65229207769409095821805644027, 2.89617252801749625577047888618, 3.13002595626263879655497781263, 3.28691638143702816522427601846, 3.31158705368128230376266593473, 3.41046824069495776981128761145, 3.57754217924805269470057020107, 3.60378502237672287487362315844, 3.82507306757703508597721357712, 3.91624214104102468778459677922
Plot not available for L-functions of degree greater than 10.
