| L(s) = 1 | + 3-s + 4-s + 3·5-s − 2·9-s + 12-s + 3·15-s − 3·16-s + 3·20-s + 4·25-s − 5·27-s + 5·31-s − 2·36-s + 6·37-s − 6·45-s − 3·48-s + 4·49-s − 15·53-s + 3·60-s − 7·64-s + 9·67-s − 15·71-s + 4·75-s − 9·80-s + 81-s − 15·89-s + 5·93-s − 6·97-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 1.34·5-s − 2/3·9-s + 0.288·12-s + 0.774·15-s − 3/4·16-s + 0.670·20-s + 4/5·25-s − 0.962·27-s + 0.898·31-s − 1/3·36-s + 0.986·37-s − 0.894·45-s − 0.433·48-s + 4/7·49-s − 2.06·53-s + 0.387·60-s − 7/8·64-s + 1.09·67-s − 1.78·71-s + 0.461·75-s − 1.00·80-s + 1/9·81-s − 1.58·89-s + 0.518·93-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.868536700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.868536700\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 91 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 101 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.61804269149265698780892668657, −9.882401166416226415873153865669, −9.590601188009767510762478021927, −9.088612380213389799186881556024, −8.527871540816042242216556571634, −8.003754542377893610921486397718, −7.33255041163428354031584364016, −6.63194036733281711018312364189, −6.15394990959925820699910873782, −5.68591232663550540863181236433, −4.93041013461069892960488838985, −4.14295500345574971300811236329, −2.99719913411464429328347298085, −2.55008190051617150431531328373, −1.69497144263362379198334245522,
1.69497144263362379198334245522, 2.55008190051617150431531328373, 2.99719913411464429328347298085, 4.14295500345574971300811236329, 4.93041013461069892960488838985, 5.68591232663550540863181236433, 6.15394990959925820699910873782, 6.63194036733281711018312364189, 7.33255041163428354031584364016, 8.003754542377893610921486397718, 8.527871540816042242216556571634, 9.088612380213389799186881556024, 9.590601188009767510762478021927, 9.882401166416226415873153865669, 10.61804269149265698780892668657
