| L(s) = 1 | + 3-s − 4-s − 4·5-s + 7-s − 9-s + 5·11-s − 12-s − 4·15-s − 3·16-s − 4·17-s + 13·19-s + 4·20-s + 21-s + 7·23-s + 6·25-s − 28-s − 2·29-s + 31-s + 5·33-s − 4·35-s + 36-s + 9·37-s − 6·41-s + 8·43-s − 5·44-s + 4·45-s − 47-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/2·4-s − 1.78·5-s + 0.377·7-s − 1/3·9-s + 1.50·11-s − 0.288·12-s − 1.03·15-s − 3/4·16-s − 0.970·17-s + 2.98·19-s + 0.894·20-s + 0.218·21-s + 1.45·23-s + 6/5·25-s − 0.188·28-s − 0.371·29-s + 0.179·31-s + 0.870·33-s − 0.676·35-s + 1/6·36-s + 1.47·37-s − 0.937·41-s + 1.21·43-s − 0.753·44-s + 0.596·45-s − 0.145·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 583009 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 583009 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.569289981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.569289981\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 8 T + p T^{2} ) \) |
| 2251 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 16 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 50 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - T - 52 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 11 T + 160 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
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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
−12.3306050126, −12.0379589628, −11.6805751131, −11.3077085728, −11.2312399413, −10.7927630510, −9.83518549114, −9.47417484434, −9.26693368433, −8.83522305040, −8.44944141105, −7.94432857806, −7.65372195732, −7.13665047045, −6.84775160667, −6.31406342663, −5.41698872876, −5.11192470641, −4.39943189470, −4.19190988549, −3.59396385459, −3.14714865052, −2.61866938000, −1.42206583068, −0.690998779123,
0.690998779123, 1.42206583068, 2.61866938000, 3.14714865052, 3.59396385459, 4.19190988549, 4.39943189470, 5.11192470641, 5.41698872876, 6.31406342663, 6.84775160667, 7.13665047045, 7.65372195732, 7.94432857806, 8.44944141105, 8.83522305040, 9.26693368433, 9.47417484434, 9.83518549114, 10.7927630510, 11.2312399413, 11.3077085728, 11.6805751131, 12.0379589628, 12.3306050126
