| L(s) = 1 | − 4-s + 6·13-s + 16-s − 2·17-s − 6·23-s + 9·25-s − 18·29-s + 14·43-s − 49-s − 6·52-s + 20·53-s + 22·61-s − 64-s + 2·68-s − 24·79-s + 6·92-s − 9·100-s + 4·101-s − 10·103-s + 4·107-s + 4·113-s + 18·116-s + 21·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.66·13-s + 1/4·16-s − 0.485·17-s − 1.25·23-s + 9/5·25-s − 3.34·29-s + 2.13·43-s − 1/7·49-s − 0.832·52-s + 2.74·53-s + 2.81·61-s − 1/8·64-s + 0.242·68-s − 2.70·79-s + 0.625·92-s − 0.899·100-s + 0.398·101-s − 0.985·103-s + 0.386·107-s + 0.376·113-s + 1.67·116-s + 1.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.902487436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.902487436\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.605415753795048473971187581684, −8.923151729380885625586443407000, −8.724349818338890100419036863488, −8.715142890245018728523541581837, −8.071395067409180812612779582792, −7.56744132373493208535869427739, −7.07276906570607267469539849406, −7.02866321813685786888235586540, −6.12366523540809836145610259230, −5.98239029814800595962653552624, −5.41766685014813799512945661417, −5.33357163356571344567471986890, −4.26293531257245778785264389355, −4.23352339657263625153491165396, −3.70749095648718184002549111569, −3.35105147874108255698222362429, −2.48381326839205881934110771994, −2.06119802347000057699424732207, −1.28109676730845854431453883218, −0.58334601001825132688012284631,
0.58334601001825132688012284631, 1.28109676730845854431453883218, 2.06119802347000057699424732207, 2.48381326839205881934110771994, 3.35105147874108255698222362429, 3.70749095648718184002549111569, 4.23352339657263625153491165396, 4.26293531257245778785264389355, 5.33357163356571344567471986890, 5.41766685014813799512945661417, 5.98239029814800595962653552624, 6.12366523540809836145610259230, 7.02866321813685786888235586540, 7.07276906570607267469539849406, 7.56744132373493208535869427739, 8.071395067409180812612779582792, 8.715142890245018728523541581837, 8.724349818338890100419036863488, 8.923151729380885625586443407000, 9.605415753795048473971187581684
