| L(s) = 1 | + 128·4-s − 1.22e3·7-s + 1.22e4·16-s + 1.37e4·19-s + 2.83e4·25-s − 1.56e5·28-s − 2.85e5·43-s + 8.81e5·49-s + 1.14e5·61-s + 1.04e6·64-s + 7.68e5·73-s + 1.75e6·76-s + 3.62e6·100-s − 1.49e7·112-s + 2.41e6·121-s + ⋯ |
| L(s) = 1 | + 2·4-s − 3.55·7-s + 3·16-s + 2·19-s + 1.81·25-s − 7.11·28-s − 3.58·43-s + 7.48·49-s + 0.502·61-s + 4·64-s + 1.97·73-s + 4·76-s + 3.62·100-s − 10.6·112-s + 1.36·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.937650125\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.937650125\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 28334 T^{2} + p^{12} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 610 T + p^{6} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2415278 T^{2} + p^{12} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 44461762 T^{2} + p^{12} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 128700322 T^{2} + p^{12} T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 142630 T + p^{6} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 15910908158 T^{2} + p^{12} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 57062 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 384050 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 625348114162 T^{2} + p^{12} T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
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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
−11.76801750681482580681951915627, −11.67824339855700651342988284668, −10.78749743105494916179739608221, −10.34038824062500299082952023379, −9.866268386017175872951737384725, −9.708372615221282121268120091133, −9.051593667055160841690160782559, −8.244727347374948610268563529434, −7.43437145835820032836935437453, −6.89590448765747306227971577055, −6.70156968621982486165502239887, −6.40907091800011216470061545915, −5.71127423798024978609411509631, −5.17404516588272737563776902941, −3.51621367674706951503338690643, −3.34760380671788656997649049921, −3.00342369089889293197046369813, −2.35981123181432475814092553553, −1.22326944067263947508047381993, −0.50187195318311949110212437387,
0.50187195318311949110212437387, 1.22326944067263947508047381993, 2.35981123181432475814092553553, 3.00342369089889293197046369813, 3.34760380671788656997649049921, 3.51621367674706951503338690643, 5.17404516588272737563776902941, 5.71127423798024978609411509631, 6.40907091800011216470061545915, 6.70156968621982486165502239887, 6.89590448765747306227971577055, 7.43437145835820032836935437453, 8.244727347374948610268563529434, 9.051593667055160841690160782559, 9.708372615221282121268120091133, 9.866268386017175872951737384725, 10.34038824062500299082952023379, 10.78749743105494916179739608221, 11.67824339855700651342988284668, 11.76801750681482580681951915627
