| L(s) = 1 | + 2-s + 2·3-s + 3·5-s + 2·6-s − 2·7-s − 8-s + 3·9-s + 3·10-s + 2·11-s + 4·13-s − 2·14-s + 6·15-s − 16-s + 3·18-s − 7·19-s − 4·21-s + 2·22-s − 2·24-s + 5·25-s + 4·26-s + 10·27-s + 6·29-s + 6·30-s − 8·31-s + 4·33-s − 6·35-s − 2·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1.34·5-s + 0.816·6-s − 0.755·7-s − 0.353·8-s + 9-s + 0.948·10-s + 0.603·11-s + 1.10·13-s − 0.534·14-s + 1.54·15-s − 1/4·16-s + 0.707·18-s − 1.60·19-s − 0.872·21-s + 0.426·22-s − 0.408·24-s + 25-s + 0.784·26-s + 1.92·27-s + 1.11·29-s + 1.09·30-s − 1.43·31-s + 0.696·33-s − 1.01·35-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.289560533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.289560533\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
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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
−11.25664942521452835462549044770, −10.98353865382120255724170948324, −10.43676169019491811916616801470, −9.877362423057999865143123180099, −9.736260757540450817237931022684, −8.924686139704274987518687740663, −8.904507211767582950778476744323, −8.467031651044092260009441276899, −7.84382001674572726484910320123, −7.03620393887706903165464499772, −6.59567123484551413667535859853, −6.11317790269323866411209143952, −6.03595373463731358190879665573, −4.95985926861868640547139133272, −4.64185985594449241551876473203, −3.75231918841972684403261549152, −3.51764230267867490351764021950, −2.72105439189372660242162823600, −2.15524718829239426754884768100, −1.33074000930363262732815613750,
1.33074000930363262732815613750, 2.15524718829239426754884768100, 2.72105439189372660242162823600, 3.51764230267867490351764021950, 3.75231918841972684403261549152, 4.64185985594449241551876473203, 4.95985926861868640547139133272, 6.03595373463731358190879665573, 6.11317790269323866411209143952, 6.59567123484551413667535859853, 7.03620393887706903165464499772, 7.84382001674572726484910320123, 8.467031651044092260009441276899, 8.904507211767582950778476744323, 8.924686139704274987518687740663, 9.736260757540450817237931022684, 9.877362423057999865143123180099, 10.43676169019491811916616801470, 10.98353865382120255724170948324, 11.25664942521452835462549044770
