| L(s) = 1 | + 3·4-s − 72·13-s + 41·16-s − 216·17-s − 120·23-s + 228·25-s + 48·29-s + 1.06e3·43-s + 328·49-s − 216·52-s + 864·53-s − 2.28e3·61-s + 411·64-s − 648·68-s + 288·79-s − 360·92-s + 684·100-s − 528·101-s − 2.96e3·103-s + 3.24e3·107-s − 1.41e3·113-s + 144·116-s + 4.89e3·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 3/8·4-s − 1.53·13-s + 0.640·16-s − 3.08·17-s − 1.08·23-s + 1.82·25-s + 0.307·29-s + 3.77·43-s + 0.956·49-s − 0.576·52-s + 2.23·53-s − 4.78·61-s + 0.802·64-s − 1.15·68-s + 0.410·79-s − 0.407·92-s + 0.683·100-s − 0.520·101-s − 2.83·103-s + 2.92·107-s − 1.17·113-s + 0.115·116-s + 3.67·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.708327761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.708327761\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 13 | $D_{4}$ | \( 1 + 72 T + 238 p T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 3 T^{2} - p^{5} T^{4} - 3 p^{6} T^{6} + p^{12} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 - 228 T^{2} + 33862 T^{4} - 228 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 - 328 T^{2} + 51918 T^{4} - 328 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 4896 T^{2} + 9533230 T^{4} - 4896 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $C_2$ | \( ( 1 + 54 T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 15880 T^{2} + 155242878 T^{4} - 15880 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 60 T + 1870 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 24 T + 25558 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 102136 T^{2} + 4357504590 T^{4} - 102136 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 2164 T^{2} - 2618990922 T^{4} - 2164 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 230292 T^{2} + 22372016182 T^{4} - 230292 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 532 T + 206406 T^{2} - 532 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 85056 T^{2} + 23167872958 T^{4} + 85056 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 432 T + 134134 T^{2} - 432 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 716592 T^{2} + 212420895502 T^{4} - 716592 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 1140 T + 12598 p T^{2} + 1140 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 475384 T^{2} + 105182398878 T^{4} - 475384 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 1283952 T^{2} + 666179277502 T^{4} - 1283952 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 762916 T^{2} + 359882924838 T^{4} - 762916 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 144 T + 825118 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 1633872 T^{2} + 191792062 p^{2} T^{4} - 1633872 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 2759412 T^{2} + 2896886558902 T^{4} - 2759412 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 + 53564 T^{2} + 1619815156998 T^{4} + 53564 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−9.496562976691881982089466116712, −9.222952740416132511091501245603, −8.970739713369947696623887366944, −8.615967190490185903399101696328, −8.263036900372385501786206717755, −8.257761145054209360831077569957, −7.40912475143256646982549569631, −7.34696308426532355518622749230, −7.30500279823175762453336620302, −6.87213067379417310450204450260, −6.58069158454424595187403275761, −6.03209058916779068237053169881, −5.83292012781761417997701667359, −5.78661490425240969965153329397, −4.97310403215082022596600585596, −4.59278120986566415526110661767, −4.55803331966515885335514194982, −4.21499655669178767476326044704, −3.70510994033716105615211728630, −2.97540791515606460855049990846, −2.70537249330480534237819096703, −2.21358440604302909228279161300, −2.06578689644116213817655823549, −1.04219626030381770724059870583, −0.37285511647387712587146729768,
0.37285511647387712587146729768, 1.04219626030381770724059870583, 2.06578689644116213817655823549, 2.21358440604302909228279161300, 2.70537249330480534237819096703, 2.97540791515606460855049990846, 3.70510994033716105615211728630, 4.21499655669178767476326044704, 4.55803331966515885335514194982, 4.59278120986566415526110661767, 4.97310403215082022596600585596, 5.78661490425240969965153329397, 5.83292012781761417997701667359, 6.03209058916779068237053169881, 6.58069158454424595187403275761, 6.87213067379417310450204450260, 7.30500279823175762453336620302, 7.34696308426532355518622749230, 7.40912475143256646982549569631, 8.257761145054209360831077569957, 8.263036900372385501786206717755, 8.615967190490185903399101696328, 8.970739713369947696623887366944, 9.222952740416132511091501245603, 9.496562976691881982089466116712
