| L(s) = 1 | − 2·3-s + 2·4-s + 2·7-s + 9-s − 4·11-s − 4·12-s + 12·13-s + 4·16-s − 4·17-s + 2·19-s − 4·21-s − 4·23-s + 2·27-s + 4·28-s + 16·29-s + 6·31-s + 8·33-s + 2·36-s − 14·37-s − 24·39-s − 8·41-s + 36·43-s − 8·44-s − 4·47-s − 8·48-s + 7·49-s + 8·51-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 4-s + 0.755·7-s + 1/3·9-s − 1.20·11-s − 1.15·12-s + 3.32·13-s + 16-s − 0.970·17-s + 0.458·19-s − 0.872·21-s − 0.834·23-s + 0.384·27-s + 0.755·28-s + 2.97·29-s + 1.07·31-s + 1.39·33-s + 1/3·36-s − 2.30·37-s − 3.84·39-s − 1.24·41-s + 5.48·43-s − 1.20·44-s − 0.583·47-s − 1.15·48-s + 49-s + 1.12·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.241681974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.241681974\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T - 8 T^{2} + 8 T^{3} + 279 T^{4} + 8 p T^{5} - 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 6 T + 33 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 4 T - 4 T^{2} - 56 T^{3} - 161 T^{4} - 56 p T^{5} - 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 2 T - 3 T^{2} + 62 T^{3} - 388 T^{4} + 62 p T^{5} - 3 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 4 T - 16 T^{2} - 56 T^{3} + 127 T^{4} - 56 p T^{5} - 16 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 8 T + 56 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 6 T - 27 T^{2} - 6 T^{3} + 1892 T^{4} - 6 p T^{5} - 27 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 14 T + 75 T^{2} + 658 T^{3} + 6020 T^{4} + 658 p T^{5} + 75 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 18 T + 165 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 4 T + 46 T^{2} - 496 T^{3} - 2061 T^{4} - 496 p T^{5} + 46 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 8 T - 50 T^{2} - 64 T^{3} + 6795 T^{4} - 64 p T^{5} - 50 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 116 T^{2} + 9975 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 8 T - 2 T^{2} - 448 T^{3} - 3269 T^{4} - 448 p T^{5} - 2 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 2 T + 31 T^{2} - 322 T^{3} - 4028 T^{4} - 322 p T^{5} + 31 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T + 144 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 10 T - 21 T^{2} - 250 T^{3} + 2012 T^{4} - 250 p T^{5} - 21 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 2 T - 123 T^{2} - 62 T^{3} + 9572 T^{4} - 62 p T^{5} - 123 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 8 T + 180 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 16 T + 32 T^{2} - 736 T^{3} + 19471 T^{4} - 736 p T^{5} + 32 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 250 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.59680816073005120190443971702, −7.50915357451069729783977342355, −7.44011287278583518695134941053, −7.05558052780846275197310253356, −7.00456498950766796807564399515, −6.46368048800085617685875886813, −6.17991254443848803928791093872, −5.95355349157480781183110511691, −5.92277023270848668495261901851, −5.88870260812422690972763569435, −5.75556878130513889407475945658, −4.90219962096394044195593052436, −4.76185966475079827734485526396, −4.67672434575019028170946460263, −4.62101962538669186801178569230, −3.80927029638126628561973823840, −3.71232974096582585921858578820, −3.44207219437497943693608926365, −3.12751126942958120730267843995, −2.56817056252267954857911553392, −2.35162858068350172181695472222, −2.08236511203292890354092003344, −1.42435431579097682436193401469, −0.940784216866663612608743471700, −0.839952529568944702235911553116,
0.839952529568944702235911553116, 0.940784216866663612608743471700, 1.42435431579097682436193401469, 2.08236511203292890354092003344, 2.35162858068350172181695472222, 2.56817056252267954857911553392, 3.12751126942958120730267843995, 3.44207219437497943693608926365, 3.71232974096582585921858578820, 3.80927029638126628561973823840, 4.62101962538669186801178569230, 4.67672434575019028170946460263, 4.76185966475079827734485526396, 4.90219962096394044195593052436, 5.75556878130513889407475945658, 5.88870260812422690972763569435, 5.92277023270848668495261901851, 5.95355349157480781183110511691, 6.17991254443848803928791093872, 6.46368048800085617685875886813, 7.00456498950766796807564399515, 7.05558052780846275197310253356, 7.44011287278583518695134941053, 7.50915357451069729783977342355, 7.59680816073005120190443971702
