| L(s) = 1 | + 4·3-s + 6·9-s + 4·11-s + 4·17-s + 4·19-s + 25-s − 4·27-s + 16·33-s + 4·41-s + 12·43-s − 14·49-s + 16·51-s + 16·57-s + 12·59-s − 4·73-s + 4·75-s − 37·81-s + 24·83-s − 12·89-s + 4·97-s + 24·99-s − 4·107-s + 4·113-s − 10·121-s + 16·123-s + 127-s + 48·129-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 2·9-s + 1.20·11-s + 0.970·17-s + 0.917·19-s + 1/5·25-s − 0.769·27-s + 2.78·33-s + 0.624·41-s + 1.82·43-s − 2·49-s + 2.24·51-s + 2.11·57-s + 1.56·59-s − 0.468·73-s + 0.461·75-s − 4.11·81-s + 2.63·83-s − 1.27·89-s + 0.406·97-s + 2.41·99-s − 0.386·107-s + 0.376·113-s − 0.909·121-s + 1.44·123-s + 0.0887·127-s + 4.22·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2163200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2163200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.223874108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.223874108\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−7.76319041513224922753287966748, −7.44095337164983387150060505996, −7.22602053082602449735325415510, −6.33588991532526860429499225622, −6.28727870819603758957301866812, −5.43479364046733402754835461064, −5.24172374409154567149889570954, −4.38796175153128331239805240266, −3.83609522350488929158971696987, −3.73594560911583979412114290925, −2.95784688889706959154847700703, −2.93871105322865322021254198939, −2.18201972761241363090254415749, −1.63546887865200897104772810976, −0.910977372178463170757222920731,
0.910977372178463170757222920731, 1.63546887865200897104772810976, 2.18201972761241363090254415749, 2.93871105322865322021254198939, 2.95784688889706959154847700703, 3.73594560911583979412114290925, 3.83609522350488929158971696987, 4.38796175153128331239805240266, 5.24172374409154567149889570954, 5.43479364046733402754835461064, 6.28727870819603758957301866812, 6.33588991532526860429499225622, 7.22602053082602449735325415510, 7.44095337164983387150060505996, 7.76319041513224922753287966748
