| L(s) = 1 | + 4·3-s + 10·9-s − 6·13-s − 8·17-s − 10·23-s + 7·25-s + 20·27-s + 22·29-s − 24·39-s − 14·43-s − 2·49-s − 32·51-s + 6·53-s − 28·61-s − 40·69-s + 28·75-s − 10·79-s + 35·81-s + 88·87-s + 12·101-s + 40·103-s + 18·113-s − 60·117-s + 8·121-s + 127-s − 56·129-s + 131-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 10/3·9-s − 1.66·13-s − 1.94·17-s − 2.08·23-s + 7/5·25-s + 3.84·27-s + 4.08·29-s − 3.84·39-s − 2.13·43-s − 2/7·49-s − 4.48·51-s + 0.824·53-s − 3.58·61-s − 4.81·69-s + 3.23·75-s − 1.12·79-s + 35/9·81-s + 9.43·87-s + 1.19·101-s + 3.94·103-s + 1.69·113-s − 5.54·117-s + 8/11·121-s + 0.0887·127-s − 4.93·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.643609582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.643609582\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $C_2^3$ | \( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 190 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 19 | $D_4\times C_2$ | \( 1 - 67 T^{2} + 1840 T^{4} - 67 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 11 T + 84 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 115 T^{2} + 5224 T^{4} - 115 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 7 T + 94 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 167 T^{2} + 11284 T^{4} - 167 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 3 T + 104 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 40 T^{2} - 866 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 14 T + 154 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 72 T^{2} + 4766 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 40 T^{2} + 4974 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 139 T^{2} + 14260 T^{4} - 139 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 5 T + 126 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 255 T^{2} + 29996 T^{4} - 255 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 279 T^{2} + 35264 T^{4} - 279 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 131 T^{2} + 9300 T^{4} - 131 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.04734015611780011267446586634, −5.83328641777180902927745812578, −5.47045775723888958076619982971, −4.90766571342073407627527751261, −4.89904964903030501289331512964, −4.88590226391969724990337670359, −4.72186530590135046632725400104, −4.67851021375844295226796764866, −4.20790226748482931271047003550, −4.05189332631339927802187754106, −4.04736062085236287314641351038, −3.60632978721397702721242191279, −3.35173413351323149639305273456, −3.16154124545013455535212115578, −2.97223905688594642133116021390, −2.82324144808772151723102958241, −2.51803697203961704069657230666, −2.35259887168245158836968647223, −2.33848754360722487248383457312, −1.82943760961141524483583867119, −1.76628662938610644915799378518, −1.36927785300550973059197725598, −1.19694384613843280105982690085, −0.50331929334794091690252944687, −0.38251575819903044128807983553,
0.38251575819903044128807983553, 0.50331929334794091690252944687, 1.19694384613843280105982690085, 1.36927785300550973059197725598, 1.76628662938610644915799378518, 1.82943760961141524483583867119, 2.33848754360722487248383457312, 2.35259887168245158836968647223, 2.51803697203961704069657230666, 2.82324144808772151723102958241, 2.97223905688594642133116021390, 3.16154124545013455535212115578, 3.35173413351323149639305273456, 3.60632978721397702721242191279, 4.04736062085236287314641351038, 4.05189332631339927802187754106, 4.20790226748482931271047003550, 4.67851021375844295226796764866, 4.72186530590135046632725400104, 4.88590226391969724990337670359, 4.89904964903030501289331512964, 4.90766571342073407627527751261, 5.47045775723888958076619982971, 5.83328641777180902927745812578, 6.04734015611780011267446586634
