L(s) = 1 | + 108·9-s − 1.63e3·29-s − 5.32e3·41-s − 1.09e4·49-s − 1.51e4·61-s + 7.29e3·81-s + 4.72e4·89-s − 6.85e4·101-s + 5.02e4·109-s + 5.66e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.23e5·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
L(s) = 1 | + 4/3·9-s − 1.94·29-s − 3.16·41-s − 4.57·49-s − 4.07·61-s + 10/9·81-s + 5.96·89-s − 6.72·101-s + 4.22·109-s + 3.86·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 7.80·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(42.43314976\) |
\(L(\frac12)\) |
\(\approx\) |
\(42.43314976\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 5 | \( 1 \) |
good | 7 | \( ( 1 + 5494 T^{2} + 15936267 T^{4} + 5494 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 11 | \( ( 1 - 28300 T^{2} + 562066662 T^{4} - 28300 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 13 | \( ( 1 - 55753 T^{2} + p^{8} T^{4} )^{4} \) |
| 17 | \( ( 1 - 306364 T^{2} + 37245043590 T^{4} - 306364 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 19 | \( ( 1 + 16346 T^{2} + 26496359067 T^{4} + 16346 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 23 | \( ( 1 + 867052 T^{2} + 344181571302 T^{4} + 867052 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 29 | \( ( 1 + 408 T + 671342 T^{2} + 408 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 31 | \( ( 1 - 1072150 T^{2} + 381333356907 T^{4} - 1072150 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 37 | \( ( 1 - 6906140 T^{2} + 18928093199238 T^{4} - 6906140 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 41 | \( ( 1 + 1332 T + 2189474 T^{2} + 1332 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 43 | \( ( 1 + 12946294 T^{2} + 65145308437707 T^{4} + 12946294 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 47 | \( ( 1 + 15422764 T^{2} + 106752772391142 T^{4} + 15422764 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 53 | \( ( 1 - 28369012 T^{2} + 324473585653158 T^{4} - 28369012 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 59 | \( ( 1 - 24263020 T^{2} + 430241364354342 T^{4} - 24263020 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 61 | \( ( 1 + 3794 T + 21463587 T^{2} + 3794 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 67 | \( ( 1 + 55586134 T^{2} + 1434114297172587 T^{4} + 55586134 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 71 | \( ( 1 - 89052244 T^{2} + 3270316864167462 T^{4} - 89052244 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 73 | \( ( 1 - 94409372 T^{2} + 3775173347372358 T^{4} - 94409372 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 79 | \( ( 1 - 131837476 T^{2} + 7248974063083590 T^{4} - 131837476 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 83 | \( ( 1 + 99346348 T^{2} + 4925474293745958 T^{4} + 99346348 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 89 | \( ( 1 - 11808 T + 130620098 T^{2} - 11808 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 97 | \( ( 1 - 175161746 T^{2} + 16599386047101651 T^{4} - 175161746 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
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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
−3.47288702983014651975829158009, −3.35147504138416672353364249665, −3.16620285065863581160673866313, −3.13499760768557170742295083630, −3.10186597572788438496555668118, −3.09189796197323961109631235564, −2.92399877081986087198483205308, −2.73858161058104540864984572936, −2.47743818236416814705849935802, −2.29367986486440448667670094424, −1.95835608863673513722985915563, −1.95588852345726228366304738375, −1.79438901927917650405737844683, −1.78939508502645529823612621873, −1.70403503230066871073386402746, −1.59434363348483224190456662274, −1.57201623291151642664357279966, −1.33205117986161981810342708897, −1.09699982209548505020048850797, −0.74069329310613980639540826360, −0.51353938625577106320724896787, −0.47504058868127507963990917213, −0.42918677125548684262852142015, −0.40911985531046607579952068829, −0.35071917894099382031106338522,
0.35071917894099382031106338522, 0.40911985531046607579952068829, 0.42918677125548684262852142015, 0.47504058868127507963990917213, 0.51353938625577106320724896787, 0.74069329310613980639540826360, 1.09699982209548505020048850797, 1.33205117986161981810342708897, 1.57201623291151642664357279966, 1.59434363348483224190456662274, 1.70403503230066871073386402746, 1.78939508502645529823612621873, 1.79438901927917650405737844683, 1.95588852345726228366304738375, 1.95835608863673513722985915563, 2.29367986486440448667670094424, 2.47743818236416814705849935802, 2.73858161058104540864984572936, 2.92399877081986087198483205308, 3.09189796197323961109631235564, 3.10186597572788438496555668118, 3.13499760768557170742295083630, 3.16620285065863581160673866313, 3.35147504138416672353364249665, 3.47288702983014651975829158009
Plot not available for L-functions of degree greater than 10.