| L(s) = 1 | − 208·31-s + 4.91e3·61-s − 1.14e3·81-s + 9.38e3·121-s + 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 + ⋯ |
| L(s) = 1 | − 1.20·31-s + 10.3·61-s − 1.56·81-s + 7.05·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + 0.000281·233-s + 0.000270·239-s + 0.000267·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(12.76624945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.76624945\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 127 p^{2} T^{4} + p^{12} T^{8} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 98786 T^{4} + p^{12} T^{8} )^{2} \) |
| 11 | \( ( 1 - 2347 T^{2} + p^{6} T^{4} )^{4} \) |
| 13 | \( ( 1 - 7905934 T^{4} + p^{12} T^{8} )^{2} \) |
| 17 | \( ( 1 - 46530097 T^{4} + p^{12} T^{8} )^{2} \) |
| 19 | \( ( 1 - 11509 T^{2} + p^{6} T^{4} )^{4} \) |
| 23 | \( ( 1 + 126395138 T^{4} + p^{12} T^{8} )^{2} \) |
| 29 | \( ( 1 - 31862 T^{2} + p^{6} T^{4} )^{4} \) |
| 31 | \( ( 1 + 26 T + p^{3} T^{2} )^{8} \) |
| 37 | \( ( 1 - 5127712462 T^{4} + p^{12} T^{8} )^{2} \) |
| 41 | \( ( 1 + 28793 T^{2} + p^{6} T^{4} )^{4} \) |
| 43 | \( ( 1 + 1102960466 T^{4} + p^{12} T^{8} )^{2} \) |
| 47 | \( ( 1 - 20383723582 T^{4} + p^{12} T^{8} )^{2} \) |
| 53 | \( ( 1 + 42091990418 T^{4} + p^{12} T^{8} )^{2} \) |
| 59 | \( ( 1 - 44102 T^{2} + p^{6} T^{4} )^{4} \) |
| 61 | \( ( 1 - 614 T + p^{3} T^{2} )^{8} \) |
| 67 | \( ( 1 + 114543970631 T^{4} + p^{12} T^{8} )^{2} \) |
| 71 | \( ( 1 - 635182 T^{2} + p^{6} T^{4} )^{4} \) |
| 73 | \( ( 1 - 241999343857 T^{4} + p^{12} T^{8} )^{2} \) |
| 79 | \( ( 1 - 642682 T^{2} + p^{6} T^{4} )^{4} \) |
| 83 | \( ( 1 - 9988446697 T^{4} + p^{12} T^{8} )^{2} \) |
| 89 | \( ( 1 + 237823 T^{2} + p^{6} T^{4} )^{4} \) |
| 97 | \( ( 1 - 907201970494 T^{4} + p^{12} T^{8} )^{2} \) |
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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
−4.75880769110959007037197912490, −4.42671279713117247312295742768, −4.37403997883740500263998150589, −4.30143599487499776883670200927, −4.24520830483177004491632697439, −4.14873728286243647078732941962, −3.63441200855151022640849184847, −3.54449088753335917016974987089, −3.52867415765701894244630183749, −3.51628607432465640600005075395, −3.10547233320278200955874907249, −3.07837731859472024004042757117, −2.86499509411886841585660217937, −2.49606729949586409804439523960, −2.32474807635313704914446047089, −2.16889874426063930377176814270, −2.06908424283894193419718925416, −1.96099839516331303697840166133, −1.71188025785266639713268798416, −1.45035952525054745691341660163, −0.906758817341735682329121403272, −0.892245779808733057875001197916, −0.67851560616824043643828252286, −0.52063874867570949627836138410, −0.31972375392114623468425829921,
0.31972375392114623468425829921, 0.52063874867570949627836138410, 0.67851560616824043643828252286, 0.892245779808733057875001197916, 0.906758817341735682329121403272, 1.45035952525054745691341660163, 1.71188025785266639713268798416, 1.96099839516331303697840166133, 2.06908424283894193419718925416, 2.16889874426063930377176814270, 2.32474807635313704914446047089, 2.49606729949586409804439523960, 2.86499509411886841585660217937, 3.07837731859472024004042757117, 3.10547233320278200955874907249, 3.51628607432465640600005075395, 3.52867415765701894244630183749, 3.54449088753335917016974987089, 3.63441200855151022640849184847, 4.14873728286243647078732941962, 4.24520830483177004491632697439, 4.30143599487499776883670200927, 4.37403997883740500263998150589, 4.42671279713117247312295742768, 4.75880769110959007037197912490
Plot not available for L-functions of degree greater than 10.
