| L(s) = 1 | − 5-s + 7-s − 3·9-s + 11-s − 13-s − 4·19-s − 5·23-s − 5·25-s + 3·29-s − 31-s − 35-s − 2·37-s + 5·41-s − 7·43-s + 3·45-s − 3·47-s − 9·49-s − 10·53-s − 55-s − 59-s + 7·61-s − 3·63-s + 65-s − 67-s − 4·73-s + 77-s + 9·79-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 9-s + 0.301·11-s − 0.277·13-s − 0.917·19-s − 1.04·23-s − 25-s + 0.557·29-s − 0.179·31-s − 0.169·35-s − 0.328·37-s + 0.780·41-s − 1.06·43-s + 0.447·45-s − 0.437·47-s − 9/7·49-s − 1.37·53-s − 0.134·55-s − 0.130·59-s + 0.896·61-s − 0.377·63-s + 0.124·65-s − 0.122·67-s − 0.468·73-s + 0.113·77-s + 1.01·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 46 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 7 T + 22 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 90 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + T - 66 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 7 T + 22 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 9 T + 18 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 7 T + 60 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
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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
−14.4580859175, −14.1050793907, −13.6807362928, −13.1465297764, −12.6005526589, −12.1717858913, −11.7882326680, −11.3506640762, −11.0104616604, −10.5102072918, −9.76146842052, −9.63698748698, −8.81954307993, −8.37921130387, −8.05994717068, −7.61713326587, −6.87438151773, −6.26403030326, −5.97074399381, −5.14369045581, −4.67545253481, −3.94567140916, −3.39778622448, −2.51324226833, −1.72722797106, 0,
1.72722797106, 2.51324226833, 3.39778622448, 3.94567140916, 4.67545253481, 5.14369045581, 5.97074399381, 6.26403030326, 6.87438151773, 7.61713326587, 8.05994717068, 8.37921130387, 8.81954307993, 9.63698748698, 9.76146842052, 10.5102072918, 11.0104616604, 11.3506640762, 11.7882326680, 12.1717858913, 12.6005526589, 13.1465297764, 13.6807362928, 14.1050793907, 14.4580859175
