| L(s) = 1 | + 11·7-s − 44·13-s + 37·19-s − 25·25-s + 13·31-s − 26·37-s − 122·43-s + 72·49-s + 121·61-s − 122·67-s − 143·73-s + 142·79-s − 484·91-s + 334·97-s − 194·103-s − 71·109-s − 121·121-s + 127-s + 131-s + 407·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 11/7·7-s − 3.38·13-s + 1.94·19-s − 25-s + 0.419·31-s − 0.702·37-s − 2.83·43-s + 1.46·49-s + 1.98·61-s − 1.82·67-s − 1.95·73-s + 1.79·79-s − 5.31·91-s + 3.44·97-s − 1.88·103-s − 0.651·109-s − 121-s + 0.00787·127-s + 0.00763·131-s + 3.06·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.743733758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.743733758\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 - 11 T + p^{2} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )( 1 - 11 T + p^{2} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 59 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )( 1 + 73 T + p^{2} T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 61 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )( 1 - 47 T + p^{2} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 13 T + p^{2} T^{2} )( 1 + 109 T + p^{2} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )( 1 + 97 T + p^{2} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 131 T + p^{2} T^{2} )( 1 - 11 T + p^{2} T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 167 T + p^{2} T^{2} )^{2} \) |
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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
−10.18180398838282078440602391957, −9.926258634754627123060549696431, −9.695009576894335720349513274368, −9.116348041821956451485137785688, −8.637015580607987953422656196747, −8.007935522325443373220525499277, −7.77675929563331017016381177584, −7.35554056574911177869694079634, −7.13858293028610247451920256356, −6.54983830522572604608383475915, −5.69940630553941643181161136372, −5.19138990918447882201297068147, −4.98700571114432381620092994309, −4.74309213082229478332193847537, −4.00933821217462990339506873754, −3.22112682922620523626041015282, −2.68019878585152872173924439207, −2.02133196854073560470015935113, −1.56330464382459191433941866165, −0.44599787284632263654718859506,
0.44599787284632263654718859506, 1.56330464382459191433941866165, 2.02133196854073560470015935113, 2.68019878585152872173924439207, 3.22112682922620523626041015282, 4.00933821217462990339506873754, 4.74309213082229478332193847537, 4.98700571114432381620092994309, 5.19138990918447882201297068147, 5.69940630553941643181161136372, 6.54983830522572604608383475915, 7.13858293028610247451920256356, 7.35554056574911177869694079634, 7.77675929563331017016381177584, 8.007935522325443373220525499277, 8.637015580607987953422656196747, 9.116348041821956451485137785688, 9.695009576894335720349513274368, 9.926258634754627123060549696431, 10.18180398838282078440602391957
