| L(s) = 1 | + 4·3-s + 8·7-s + 8·9-s − 8·11-s − 16-s − 8·17-s + 32·21-s + 16·23-s + 12·27-s + 12·31-s − 32·33-s − 16·37-s + 20·41-s + 8·47-s − 4·48-s + 32·49-s − 32·51-s + 16·59-s + 40·61-s + 64·63-s − 8·67-s + 64·69-s + 36·71-s − 64·77-s − 24·79-s + 23·81-s − 16·83-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 3.02·7-s + 8/3·9-s − 2.41·11-s − 1/4·16-s − 1.94·17-s + 6.98·21-s + 3.33·23-s + 2.30·27-s + 2.15·31-s − 5.57·33-s − 2.63·37-s + 3.12·41-s + 1.16·47-s − 0.577·48-s + 32/7·49-s − 4.48·51-s + 2.08·59-s + 5.12·61-s + 8.06·63-s − 0.977·67-s + 7.70·69-s + 4.27·71-s − 7.29·77-s − 2.70·79-s + 23/9·81-s − 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.568912114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.568912114\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 1654 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 540 T^{3} + 3854 T^{4} - 540 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 20 T + 200 T^{2} - 1500 T^{3} + 10094 T^{4} - 1500 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 64 T^{2} + 3922 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 6706 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 312 T^{3} + 2258 T^{4} + 312 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 36 T + 648 T^{2} - 8244 T^{3} + 79918 T^{4} - 8244 p T^{5} + 648 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^3$ | \( 1 - 8158 T^{4} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 12 T + 186 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 1584 T^{3} + 19346 T^{4} + 1584 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 516 T^{3} + 1582 T^{4} + 516 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 40 T + 800 T^{2} + 11720 T^{3} + 133282 T^{4} + 11720 p T^{5} + 800 p^{2} T^{6} + 40 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.264786965042379272446274953611, −8.185400764787085542435244941696, −7.86880069068746052936271654336, −7.66654562177417428572022241087, −7.13197788453849232604205647059, −7.06214038231092831265208562953, −6.94476538968564153935071855363, −6.85225655755882093286751277852, −6.31214206160717987261306587479, −5.47834947827813058769572161186, −5.44627969770965224977002386250, −5.35242605508780489440065683198, −5.02840338038339822089732637621, −4.89159059447817598370733025184, −4.41861980167925871972691503024, −4.22523757950332512849220004121, −3.88605592448865906594288773242, −3.75804137270984099687256149983, −2.81618196515211359684429833943, −2.66963482358206223592441688381, −2.55084595636004700514206803790, −2.49752041749616048376467577338, −2.03219767794471163105800658191, −1.28273881638284079151917126003, −1.05054651183583015538093518715,
1.05054651183583015538093518715, 1.28273881638284079151917126003, 2.03219767794471163105800658191, 2.49752041749616048376467577338, 2.55084595636004700514206803790, 2.66963482358206223592441688381, 2.81618196515211359684429833943, 3.75804137270984099687256149983, 3.88605592448865906594288773242, 4.22523757950332512849220004121, 4.41861980167925871972691503024, 4.89159059447817598370733025184, 5.02840338038339822089732637621, 5.35242605508780489440065683198, 5.44627969770965224977002386250, 5.47834947827813058769572161186, 6.31214206160717987261306587479, 6.85225655755882093286751277852, 6.94476538968564153935071855363, 7.06214038231092831265208562953, 7.13197788453849232604205647059, 7.66654562177417428572022241087, 7.86880069068746052936271654336, 8.185400764787085542435244941696, 8.264786965042379272446274953611
