| L(s) = 1 | + 2·7-s − 12·19-s + 6·25-s − 8·29-s + 12·37-s + 8·47-s + 6·49-s + 16·53-s + 16·59-s − 8·83-s − 32·103-s − 16·113-s + 34·121-s + 127-s + 131-s − 24·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 12·169-s + 173-s + 12·175-s + 179-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 2.75·19-s + 6/5·25-s − 1.48·29-s + 1.97·37-s + 1.16·47-s + 6/7·49-s + 2.19·53-s + 2.08·59-s − 0.878·83-s − 3.15·103-s − 1.50·113-s + 3.09·121-s + 0.0887·127-s + 0.0873·131-s − 2.08·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.923·169-s + 0.0760·173-s + 0.907·175-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9439027850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9439027850\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 6 T^{2} + 42 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 34 T^{2} + 514 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 12 T^{2} + 102 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 22 T^{2} + 682 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 82 T^{2} + 2722 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $D_{4}$ | \( ( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 102 T^{2} + 5130 T^{4} - 102 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 - 24 T^{2} + 3774 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 132 T^{2} + 10710 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 144 T^{2} + 10830 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 114 T^{2} + 8418 T^{4} - 114 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 236 T^{2} + 24310 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 288 T^{2} + 33150 T^{4} - 288 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 294 T^{2} + 36618 T^{4} - 294 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 204 T^{2} + 28950 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.91612143470245861302994207363, −5.71099643029847882081901224974, −5.48146013074147324339208885807, −5.47846367953164524373357654646, −5.43852326401037566459308099573, −4.72636724881449128168885727011, −4.70893279463178414689330424258, −4.63353129081826205525162296349, −4.36516226553020143976959940763, −4.13731957043879607625317647919, −3.97678900870088521952036143642, −3.95697952201924163589771747619, −3.47437646706531176724851991023, −3.42134106462095193158855587404, −2.90405524661069436842178076853, −2.83788718755514118861480501929, −2.58928783465127522574550550143, −2.21089357234790586987892699902, −2.09759799297850389305207675311, −1.94840963133941337043891174465, −1.80155739626348149836074707189, −1.05924310513774597501953971531, −0.896122747319347707841346595754, −0.889226535983573163484876571120, −0.13246836006065582480695438551,
0.13246836006065582480695438551, 0.889226535983573163484876571120, 0.896122747319347707841346595754, 1.05924310513774597501953971531, 1.80155739626348149836074707189, 1.94840963133941337043891174465, 2.09759799297850389305207675311, 2.21089357234790586987892699902, 2.58928783465127522574550550143, 2.83788718755514118861480501929, 2.90405524661069436842178076853, 3.42134106462095193158855587404, 3.47437646706531176724851991023, 3.95697952201924163589771747619, 3.97678900870088521952036143642, 4.13731957043879607625317647919, 4.36516226553020143976959940763, 4.63353129081826205525162296349, 4.70893279463178414689330424258, 4.72636724881449128168885727011, 5.43852326401037566459308099573, 5.47846367953164524373357654646, 5.48146013074147324339208885807, 5.71099643029847882081901224974, 5.91612143470245861302994207363
