| L(s) = 1 | + 2·3-s − 3·4-s + 6·7-s + 2·8-s − 2·9-s + 2·11-s − 6·12-s + 14·13-s + 4·16-s + 14·17-s − 4·19-s + 12·21-s − 4·23-s + 4·24-s − 4·27-s − 18·28-s + 4·29-s − 8·32-s + 4·33-s + 6·36-s + 2·37-s + 28·39-s − 8·41-s + 4·43-s − 6·44-s + 2·47-s + 8·48-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 3/2·4-s + 2.26·7-s + 0.707·8-s − 2/3·9-s + 0.603·11-s − 1.73·12-s + 3.88·13-s + 16-s + 3.39·17-s − 0.917·19-s + 2.61·21-s − 0.834·23-s + 0.816·24-s − 0.769·27-s − 3.40·28-s + 0.742·29-s − 1.41·32-s + 0.696·33-s + 36-s + 0.328·37-s + 4.48·39-s − 1.24·41-s + 0.609·43-s − 0.904·44-s + 0.291·47-s + 1.15·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.042434943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.042434943\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 + 3 T^{2} - p T^{3} + 5 T^{4} - p^{2} T^{5} + 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | $C_2 \wr S_4$ | \( 1 - 2 T + 2 p T^{2} - 4 p T^{3} + 23 T^{4} - 4 p^{2} T^{5} + 2 p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 6 T + 32 T^{2} - 114 T^{3} + 346 T^{4} - 114 p T^{5} + 32 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 2 T + 28 T^{2} - 2 p T^{3} + 346 T^{4} - 2 p^{2} T^{5} + 28 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 14 T + 118 T^{2} - 664 T^{3} + 2791 T^{4} - 664 p T^{5} + 118 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 14 T + 126 T^{2} - 778 T^{3} + 3726 T^{4} - 778 p T^{5} + 126 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 4 T + 70 T^{2} + 200 T^{3} + 1918 T^{4} + 200 p T^{5} + 70 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 4 T + 94 T^{2} - 344 T^{3} + 3775 T^{4} - 344 p T^{5} + 94 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 50 T^{2} - 256 T^{3} + 1011 T^{4} - 256 p T^{5} + 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 2 T + 76 T^{2} + 158 T^{3} + 2410 T^{4} + 158 p T^{5} + 76 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 8 T + 70 T^{2} + 616 T^{3} + 4863 T^{4} + 616 p T^{5} + 70 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 4 T + 54 T^{2} - 80 T^{3} + 2910 T^{4} - 80 p T^{5} + 54 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 2 T + 50 T^{2} - 116 T^{3} + 4795 T^{4} - 116 p T^{5} + 50 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 4 T + 108 T^{2} - 524 T^{3} + 8022 T^{4} - 524 p T^{5} + 108 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 216 T^{2} + 16 T^{3} + 18542 T^{4} + 16 p T^{5} + 216 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 8 T + 142 T^{2} + 316 T^{3} + 7126 T^{4} + 316 p T^{5} + 142 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 2 T + 260 T^{2} + 390 T^{3} + 25866 T^{4} + 390 p T^{5} + 260 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 24 T + 350 T^{2} + 3544 T^{3} + 32183 T^{4} + 3544 p T^{5} + 350 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 18 T + 270 T^{2} - 2088 T^{3} + 20423 T^{4} - 2088 p T^{5} + 270 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 24 T + 494 T^{2} - 6100 T^{3} + 65598 T^{4} - 6100 p T^{5} + 494 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 6 T + 62 T^{2} + 518 T^{3} + 11462 T^{4} + 518 p T^{5} + 62 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 8 T + 270 T^{2} + 1764 T^{3} + 34598 T^{4} + 1764 p T^{5} + 270 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 34 T + 736 T^{2} - 10514 T^{3} + 119290 T^{4} - 10514 p T^{5} + 736 p^{2} T^{6} - 34 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.82812307141999980071444774809, −7.75083509065728420817956679281, −7.44783128667451917948868963626, −7.42720485621817451973495191955, −6.53607280146561071499538102277, −6.49159904807375997536115669209, −6.25060654131136835163175550793, −5.95068095470485595961721833205, −5.92348658354859301297630948063, −5.30641567956163323581022170933, −5.22144918321107478993053293741, −5.10117758063551097253643011812, −4.81127330466023954237722377944, −4.28590330362798298400304013188, −4.15187581709362657885258688713, −3.95735226236595590693161379846, −3.61703720831216474199518740096, −3.32703472465273216772733312966, −3.32405971131030352518335656728, −2.78517783335880054939484483933, −2.31118076945766580989192504179, −1.56084702746934491231264930856, −1.55401650142048955815048858419, −1.27175992020449611279554014976, −0.842244458535962951617433304652,
0.842244458535962951617433304652, 1.27175992020449611279554014976, 1.55401650142048955815048858419, 1.56084702746934491231264930856, 2.31118076945766580989192504179, 2.78517783335880054939484483933, 3.32405971131030352518335656728, 3.32703472465273216772733312966, 3.61703720831216474199518740096, 3.95735226236595590693161379846, 4.15187581709362657885258688713, 4.28590330362798298400304013188, 4.81127330466023954237722377944, 5.10117758063551097253643011812, 5.22144918321107478993053293741, 5.30641567956163323581022170933, 5.92348658354859301297630948063, 5.95068095470485595961721833205, 6.25060654131136835163175550793, 6.49159904807375997536115669209, 6.53607280146561071499538102277, 7.42720485621817451973495191955, 7.44783128667451917948868963626, 7.75083509065728420817956679281, 7.82812307141999980071444774809
