| L(s) = 1 | + 16·4-s + 144·16-s − 160·19-s − 224·31-s − 56·49-s − 160·61-s + 960·64-s − 2.56e3·76-s + 224·109-s + 1.80e3·121-s − 3.58e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.69e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 896·196-s + ⋯ |
| L(s) = 1 | + 4·4-s + 9·16-s − 8.42·19-s − 7.22·31-s − 8/7·49-s − 2.62·61-s + 15·64-s − 33.6·76-s + 2.05·109-s + 14.9·121-s − 28.9·124-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 + 10.0·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s − 4.57·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4579631342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4579631342\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p T^{2} )^{8} \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 + p T^{2} )^{8} \) |
| good | 11 | \( ( 1 - 452 T^{2} + 80246 T^{4} - 452 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 13 | \( ( 1 - 848 T^{2} + 347428 T^{4} - 92436976 T^{6} + 17958310790 T^{8} - 92436976 p^{4} T^{10} + 347428 p^{8} T^{12} - 848 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 17 | \( ( 1 + 1416 T^{2} + 924316 T^{4} + 385673912 T^{6} + 122952882246 T^{8} + 385673912 p^{4} T^{10} + 924316 p^{8} T^{12} + 1416 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 19 | \( ( 1 + 40 T + 1640 T^{2} + 39240 T^{3} + 893522 T^{4} + 39240 p^{2} T^{5} + 1640 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 23 | \( ( 1 + 1824 T^{2} + 2120644 T^{4} + 1622828384 T^{6} + 989145913350 T^{8} + 1622828384 p^{4} T^{10} + 2120644 p^{8} T^{12} + 1824 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 29 | \( ( 1 - 3936 T^{2} + 8089924 T^{4} - 11011897376 T^{6} + 10780193686470 T^{8} - 11011897376 p^{4} T^{10} + 8089924 p^{8} T^{12} - 3936 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 31 | \( ( 1 + 56 T + 3544 T^{2} + 140056 T^{3} + 4852850 T^{4} + 140056 p^{2} T^{5} + 3544 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 37 | \( ( 1 - 8440 T^{2} + 33405148 T^{4} - 81543298120 T^{6} + 134336860703878 T^{8} - 81543298120 p^{4} T^{10} + 33405148 p^{8} T^{12} - 8440 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 41 | \( ( 1 - 7672 T^{2} + 31463324 T^{4} - 85348180680 T^{6} + 167315540525510 T^{8} - 85348180680 p^{4} T^{10} + 31463324 p^{8} T^{12} - 7672 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 43 | \( ( 1 - 9208 T^{2} + 36012508 T^{4} - 83899227976 T^{6} + 157065548431750 T^{8} - 83899227976 p^{4} T^{10} + 36012508 p^{8} T^{12} - 9208 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 47 | \( ( 1 + 6184 T^{2} + 20028380 T^{4} + 56156055576 T^{6} + 141910454272454 T^{8} + 56156055576 p^{4} T^{10} + 20028380 p^{8} T^{12} + 6184 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 53 | \( ( 1 + 10360 T^{2} + 63631868 T^{4} + 277444689480 T^{6} + 889296464366918 T^{8} + 277444689480 p^{4} T^{10} + 63631868 p^{8} T^{12} + 10360 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 59 | \( ( 1 - 18520 T^{2} + 175430684 T^{4} - 1054672445160 T^{6} + 4381056788172806 T^{8} - 1054672445160 p^{4} T^{10} + 175430684 p^{8} T^{12} - 18520 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 61 | \( ( 1 + 40 T + 11148 T^{2} + 417560 T^{3} + 55229638 T^{4} + 417560 p^{2} T^{5} + 11148 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 67 | \( ( 1 - 21816 T^{2} + 251662556 T^{4} - 1885383126792 T^{6} + 9995000533194246 T^{8} - 1885383126792 p^{4} T^{10} + 251662556 p^{8} T^{12} - 21816 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 71 | \( ( 1 - 16392 T^{2} + 188159164 T^{4} - 1401732081080 T^{6} + 8319601220471430 T^{8} - 1401732081080 p^{4} T^{10} + 188159164 p^{8} T^{12} - 16392 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 73 | \( ( 1 - 20432 T^{2} + 194542948 T^{4} - 1187873859184 T^{6} + 6231171192409670 T^{8} - 1187873859184 p^{4} T^{10} + 194542948 p^{8} T^{12} - 20432 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 79 | \( ( 1 + 17588 T^{2} + 149428518 T^{4} + 17588 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 83 | \( ( 1 + 49384 T^{2} + 1103437724 T^{4} + 14534924955864 T^{6} + 123316957376683910 T^{8} + 14534924955864 p^{4} T^{10} + 1103437724 p^{8} T^{12} + 49384 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 89 | \( ( 1 - 37144 T^{2} + 665612636 T^{4} - 7784028100008 T^{6} + 68907416824043846 T^{8} - 7784028100008 p^{4} T^{10} + 665612636 p^{8} T^{12} - 37144 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 97 | \( ( 1 - 29584 T^{2} + 594190564 T^{4} - 8581835526064 T^{6} + 90245023688016070 T^{8} - 8581835526064 p^{4} T^{10} + 594190564 p^{8} T^{12} - 29584 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.95295200478049039619329132319, −1.92676391482890620773024493255, −1.92112920733833116764417003197, −1.80673858188804426120579620075, −1.78546442889380614906303049032, −1.63270069059436567438691257254, −1.55871122353516951780593144753, −1.50339752813620864801340568852, −1.49872002384531700481362499020, −1.47351916552568935610673445882, −1.35189934162599180904031845437, −0.985526488237131290649774005989, −0.969011929952314174790578451572, −0.964948303483395842733652594539, −0.909038644535590427738025040495, −0.872016408930402497875285231014, −0.812333214028260789780106008860, −0.70661602345522144733574314956, −0.69884719869120372494781979863, −0.33236126899145242679438072171, −0.24940592664430470348657658008, −0.19476172534054218785287757565, −0.18035885707529725612153489778, −0.12855695618569474469924825182, −0.04471610036804915875525318386,
0.04471610036804915875525318386, 0.12855695618569474469924825182, 0.18035885707529725612153489778, 0.19476172534054218785287757565, 0.24940592664430470348657658008, 0.33236126899145242679438072171, 0.69884719869120372494781979863, 0.70661602345522144733574314956, 0.812333214028260789780106008860, 0.872016408930402497875285231014, 0.909038644535590427738025040495, 0.964948303483395842733652594539, 0.969011929952314174790578451572, 0.985526488237131290649774005989, 1.35189934162599180904031845437, 1.47351916552568935610673445882, 1.49872002384531700481362499020, 1.50339752813620864801340568852, 1.55871122353516951780593144753, 1.63270069059436567438691257254, 1.78546442889380614906303049032, 1.80673858188804426120579620075, 1.92112920733833116764417003197, 1.92676391482890620773024493255, 1.95295200478049039619329132319
Plot not available for L-functions of degree greater than 10.
