| L(s) = 1 | − 8·2-s + 24·3-s + 32·4-s + 8·5-s − 192·6-s − 132·7-s − 64·8-s + 288·9-s − 64·10-s − 120·11-s + 768·12-s + 1.05e3·14-s + 192·15-s − 64·16-s + 680·17-s − 2.30e3·18-s − 568·19-s + 256·20-s − 3.16e3·21-s + 960·22-s − 884·23-s − 1.53e3·24-s − 464·25-s + 1.94e3·27-s − 4.22e3·28-s − 3.29e3·29-s − 1.53e3·30-s + ⋯ |
| L(s) = 1 | − 2·2-s + 8/3·3-s + 2·4-s + 8/25·5-s − 5.33·6-s − 2.69·7-s − 8-s + 32/9·9-s − 0.639·10-s − 0.991·11-s + 16/3·12-s + 5.38·14-s + 0.853·15-s − 1/4·16-s + 2.35·17-s − 7.11·18-s − 1.57·19-s + 0.639·20-s − 7.18·21-s + 1.98·22-s − 1.67·23-s − 8/3·24-s − 0.742·25-s + 8/3·27-s − 5.38·28-s − 3.91·29-s − 1.70·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.049545627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.049545627\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + p^{2} T + p^{3} T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 8 p T + 32 p^{2} T^{2} - 8 p^{5} T^{3} + p^{8} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 40 p T + 48 p^{3} T^{2} - 40 p^{5} T^{3} + p^{8} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 8 T + 528 T^{2} - 26296 T^{3} + 316992 T^{4} - 26296 p^{4} T^{5} + 528 p^{8} T^{6} - 8 p^{12} T^{7} + p^{16} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 132 T + 4748 T^{2} - 30276 p T^{3} - 23533480 T^{4} - 30276 p^{5} T^{5} + 4748 p^{8} T^{6} + 132 p^{12} T^{7} + p^{16} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 120 T + 29138 T^{2} + 4404744 T^{3} + 585333602 T^{4} + 4404744 p^{4} T^{5} + 29138 p^{8} T^{6} + 120 p^{12} T^{7} + p^{16} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 98068 T^{2} + 3982378150 T^{4} - 98068 p^{8} T^{6} + p^{16} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 568 T + 161312 T^{2} + 58531832 T^{3} + 20494430818 T^{4} + 58531832 p^{4} T^{5} + 161312 p^{8} T^{6} + 568 p^{12} T^{7} + p^{16} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 884 T + 365892 T^{2} + 167287420 T^{3} + 81560428344 T^{4} + 167287420 p^{4} T^{5} + 365892 p^{8} T^{6} + 884 p^{12} T^{7} + p^{16} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 3296 T + 4188232 T^{2} + 2614770280 T^{3} + 1394120467744 T^{4} + 2614770280 p^{4} T^{5} + 4188232 p^{8} T^{6} + 3296 p^{12} T^{7} + p^{16} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 12 T + 15524 T^{2} - 311636628 T^{3} + 3857906552 T^{4} - 311636628 p^{4} T^{5} + 15524 p^{8} T^{6} - 12 p^{12} T^{7} + p^{16} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 2200 T + 1583248 T^{2} + 610239064 T^{3} - 1895709146048 T^{4} + 610239064 p^{4} T^{5} + 1583248 p^{8} T^{6} - 2200 p^{12} T^{7} + p^{16} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 1360 T + 3093618 T^{2} + 9181709776 T^{3} + 12405660647522 T^{4} + 9181709776 p^{4} T^{5} + 3093618 p^{8} T^{6} + 1360 p^{12} T^{7} + p^{16} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 1932 T + 1866312 T^{2} + 2695961100 T^{3} - 19282350117826 T^{4} + 2695961100 p^{4} T^{5} + 1866312 p^{8} T^{6} - 1932 p^{12} T^{7} + p^{16} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 1100 T + 9351430 T^{2} - 1100 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 4088 T + 8355872 T^{2} - 595519400 T^{3} - 64537360845086 T^{4} - 595519400 p^{4} T^{5} + 8355872 p^{8} T^{6} + 4088 p^{12} T^{7} + p^{16} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 868 T + 376712 T^{2} + 2943354148 T^{3} - 217510818248642 T^{4} + 2943354148 p^{4} T^{5} + 376712 p^{8} T^{6} - 868 p^{12} T^{7} + p^{16} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 10360 T + 43536600 T^{2} - 111084485200 T^{3} + 305697932132000 T^{4} - 111084485200 p^{4} T^{5} + 43536600 p^{8} T^{6} - 10360 p^{12} T^{7} + p^{16} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 11532 T + 69600948 T^{2} - 11532 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 11028 T + 31499396 T^{2} + 227484817188 T^{3} - 2145791291603656 T^{4} + 227484817188 p^{4} T^{5} + 31499396 p^{8} T^{6} - 11028 p^{12} T^{7} + p^{16} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 15048 T + 61471538 T^{2} - 295618042824 T^{3} - 3659812751309278 T^{4} - 295618042824 p^{4} T^{5} + 61471538 p^{8} T^{6} + 15048 p^{12} T^{7} + p^{16} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 3196 T + 33411972 T^{2} + 276736991260 T^{3} - 641471695435656 T^{4} + 276736991260 p^{4} T^{5} + 33411972 p^{8} T^{6} - 3196 p^{12} T^{7} + p^{16} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 15052 T + 113281352 T^{2} + 351194693940 T^{3} + 506698533174974 T^{4} + 351194693940 p^{4} T^{5} + 113281352 p^{8} T^{6} + 15052 p^{12} T^{7} + p^{16} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 1864 T + 40679056 T^{2} - 1864 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 15420 T + 60438150 T^{2} + 1170030230100 T^{3} - 16451842790850750 T^{4} + 1170030230100 p^{4} T^{5} + 60438150 p^{8} T^{6} - 15420 p^{12} T^{7} + p^{16} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.383036351076892723606613721417, −9.244950900410888203064359510015, −9.157892911699654446204665860045, −8.381655271595630347775200128537, −8.355109730932387639841165546722, −8.226018281536792440594856076378, −7.76288394502312630912630067082, −7.72424755088376785297507178958, −7.26541669946711795050243447443, −6.89964202370594001049701913977, −6.89335600621301963128942331147, −6.04722479891384568576260471477, −5.89807751630845461166009602833, −5.65178678445062495530403888634, −5.26478118982790584924856481147, −4.08412049395023204550558500348, −3.86011211694123332332076658816, −3.75293618644638081455162092493, −3.40057208467675437449620676050, −2.69463938403095028246617745744, −2.57057978693503349937478110751, −1.95964013126799914162142096717, −1.95772908699924780060321906484, −0.74514039738059967407361016626, −0.32430554003439136527459450931,
0.32430554003439136527459450931, 0.74514039738059967407361016626, 1.95772908699924780060321906484, 1.95964013126799914162142096717, 2.57057978693503349937478110751, 2.69463938403095028246617745744, 3.40057208467675437449620676050, 3.75293618644638081455162092493, 3.86011211694123332332076658816, 4.08412049395023204550558500348, 5.26478118982790584924856481147, 5.65178678445062495530403888634, 5.89807751630845461166009602833, 6.04722479891384568576260471477, 6.89335600621301963128942331147, 6.89964202370594001049701913977, 7.26541669946711795050243447443, 7.72424755088376785297507178958, 7.76288394502312630912630067082, 8.226018281536792440594856076378, 8.355109730932387639841165546722, 8.381655271595630347775200128537, 9.157892911699654446204665860045, 9.244950900410888203064359510015, 9.383036351076892723606613721417
