| L(s) = 1 | − 4·7-s − 12·11-s + 12·13-s + 7·16-s + 12·17-s + 8·19-s − 24·23-s + 4·31-s − 8·37-s + 12·41-s + 12·47-s + 8·49-s + 12·59-s + 12·61-s − 4·67-s − 24·71-s − 16·73-s + 48·77-s + 24·79-s − 9·81-s + 12·83-s + 12·89-s − 48·91-s + 16·97-s + 12·101-s + 8·109-s − 28·112-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 3.61·11-s + 3.32·13-s + 7/4·16-s + 2.91·17-s + 1.83·19-s − 5.00·23-s + 0.718·31-s − 1.31·37-s + 1.87·41-s + 1.75·47-s + 8/7·49-s + 1.56·59-s + 1.53·61-s − 0.488·67-s − 2.84·71-s − 1.87·73-s + 5.47·77-s + 2.70·79-s − 81-s + 1.31·83-s + 1.27·89-s − 5.03·91-s + 1.62·97-s + 1.19·101-s + 0.766·109-s − 2.64·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.807540267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.807540267\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^3$ | \( 1 - 7 T^{4} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 24 T^{3} + 71 T^{4} + 24 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 312 T^{3} + 1127 T^{4} + 312 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 19 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 120 T^{3} + 434 T^{4} - 120 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 29 | $D_4\times C_2$ | \( 1 - 50 T^{2} + 1443 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 168 T^{3} - 1801 T^{4} + 168 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 264 T^{3} + 2162 T^{4} + 264 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 564 T^{3} + 4382 T^{4} - 564 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 918 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 744 T^{3} + 7463 T^{4} - 744 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 146 T^{2} + 10083 T^{4} - 146 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 600 T^{3} + 4919 T^{4} - 600 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 168 T^{3} + 2903 T^{4} + 168 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 1488 T^{3} + 16898 T^{4} + 1488 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 12 T + 188 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 1176 T^{3} + 18983 T^{4} - 1176 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 1140 T^{3} + 18014 T^{4} - 1140 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1872 T^{3} + 26978 T^{4} - 1872 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.51876774585249106329379742975, −7.01654080323599540156630810920, −6.52911175362922419944835182583, −6.44175630678338536782783479497, −6.01800468716736050253902315910, −5.86141040743008582989749503797, −5.81458848845415275100130300966, −5.66998598648983638699636388976, −5.52621394140435948539206137426, −5.40218598319757503845523634120, −5.06950035039042011604087372553, −4.47081780683779482525907112829, −4.26754754771452889362425023864, −3.85897561896678995353614852844, −3.85549686932241351782942166596, −3.33136071827607472654892190404, −3.33004573518557344127321175713, −3.15759326904439791885288520911, −2.96315011127565865977148483570, −2.42795676358966380503364822196, −1.98688856879187923615538379592, −1.90577927666420020248786038183, −1.14368160292133175553338796406, −0.826537893054539771282557102296, −0.52564240722573777521416368722,
0.52564240722573777521416368722, 0.826537893054539771282557102296, 1.14368160292133175553338796406, 1.90577927666420020248786038183, 1.98688856879187923615538379592, 2.42795676358966380503364822196, 2.96315011127565865977148483570, 3.15759326904439791885288520911, 3.33004573518557344127321175713, 3.33136071827607472654892190404, 3.85549686932241351782942166596, 3.85897561896678995353614852844, 4.26754754771452889362425023864, 4.47081780683779482525907112829, 5.06950035039042011604087372553, 5.40218598319757503845523634120, 5.52621394140435948539206137426, 5.66998598648983638699636388976, 5.81458848845415275100130300966, 5.86141040743008582989749503797, 6.01800468716736050253902315910, 6.44175630678338536782783479497, 6.52911175362922419944835182583, 7.01654080323599540156630810920, 7.51876774585249106329379742975
