| L(s) = 1 | + 3-s + 4-s − 3·5-s − 3·7-s + 3·9-s + 12-s − 3·15-s − 3·16-s + 12·17-s − 5·19-s − 3·20-s − 3·21-s + 3·23-s + 5·25-s + 8·27-s − 3·28-s + 3·29-s − 3·31-s + 9·35-s + 3·36-s + 37-s + 3·41-s + 8·43-s − 9·45-s + 9·47-s − 3·48-s − 49-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s − 1.34·5-s − 1.13·7-s + 9-s + 0.288·12-s − 0.774·15-s − 3/4·16-s + 2.91·17-s − 1.14·19-s − 0.670·20-s − 0.654·21-s + 0.625·23-s + 25-s + 1.53·27-s − 0.566·28-s + 0.557·29-s − 0.538·31-s + 1.52·35-s + 1/2·36-s + 0.164·37-s + 0.468·41-s + 1.21·43-s − 1.34·45-s + 1.31·47-s − 0.433·48-s − 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.433264434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.433264434\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 229 | $C_2$ | \( 1 - 22 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 T + 26 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 3 T + 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 9 T + 74 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 21 T + 206 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 9 T + 10 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 27 T + 322 T^{2} + 27 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 9 T + 116 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−12.71851549705215564224483303689, −11.95746448427002512799898527600, −11.63505097717245054884821850734, −10.98964044140053292295412190504, −10.53018423544395945660981113360, −9.910719419912044288584399548159, −9.752114133428491674902757175557, −8.984128243074300733924212196843, −8.421359692939895163752945508757, −7.989964464431908193475803185180, −7.45755394041827486574277251131, −6.87706287387296378946151518315, −6.75889129013033161122539479911, −5.79554627366205110108603426142, −5.15596692735389545123493792333, −4.10889680013172639564716576170, −3.92372191958998938420436600905, −3.09052741937025912461505021825, −2.58406417987637782025795833979, −1.04680712885119579694605616545,
1.04680712885119579694605616545, 2.58406417987637782025795833979, 3.09052741937025912461505021825, 3.92372191958998938420436600905, 4.10889680013172639564716576170, 5.15596692735389545123493792333, 5.79554627366205110108603426142, 6.75889129013033161122539479911, 6.87706287387296378946151518315, 7.45755394041827486574277251131, 7.989964464431908193475803185180, 8.421359692939895163752945508757, 8.984128243074300733924212196843, 9.752114133428491674902757175557, 9.910719419912044288584399548159, 10.53018423544395945660981113360, 10.98964044140053292295412190504, 11.63505097717245054884821850734, 11.95746448427002512799898527600, 12.71851549705215564224483303689
