| L(s) = 1 | − 2·3-s + 3·9-s + 4·11-s − 8·17-s − 4·19-s + 6·25-s − 4·27-s − 8·33-s − 12·43-s + 6·49-s + 16·51-s + 8·57-s − 8·59-s + 8·67-s + 12·73-s − 12·75-s + 5·81-s + 4·83-s + 12·89-s − 12·97-s + 12·99-s + 12·113-s − 6·121-s + 127-s + 24·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 1.20·11-s − 1.94·17-s − 0.917·19-s + 6/5·25-s − 0.769·27-s − 1.39·33-s − 1.82·43-s + 6/7·49-s + 2.24·51-s + 1.05·57-s − 1.04·59-s + 0.977·67-s + 1.40·73-s − 1.38·75-s + 5/9·81-s + 0.439·83-s + 1.27·89-s − 1.21·97-s + 1.20·99-s + 1.12·113-s − 0.545·121-s + 0.0887·127-s + 2.11·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.348835587713841265418946651361, −7.60727245251985296194805311220, −7.09041211704160622427186245876, −6.60959484194633590966617951973, −6.43011651683893215896478775252, −6.18574954112216205884930702732, −5.26577618621189964673764136695, −4.97535844067584729794973260908, −4.46526261364155769619797015224, −4.00905551241815641124001636304, −3.49423882126371608341418447498, −2.51975969430587701278891289310, −1.90543707638619759616599745204, −1.09582465736639484353373475455, 0,
1.09582465736639484353373475455, 1.90543707638619759616599745204, 2.51975969430587701278891289310, 3.49423882126371608341418447498, 4.00905551241815641124001636304, 4.46526261364155769619797015224, 4.97535844067584729794973260908, 5.26577618621189964673764136695, 6.18574954112216205884930702732, 6.43011651683893215896478775252, 6.60959484194633590966617951973, 7.09041211704160622427186245876, 7.60727245251985296194805311220, 8.348835587713841265418946651361
