| L(s) = 1 | + 3-s + 4-s − 5-s + 9-s + 12-s − 15-s − 3·16-s − 4·17-s − 20-s − 2·25-s + 27-s + 36-s − 4·37-s + 12·41-s + 8·43-s − 45-s − 3·48-s − 7·49-s − 4·51-s − 8·59-s − 60-s − 7·64-s − 8·67-s − 4·68-s − 2·75-s + 3·80-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s − 0.447·5-s + 1/3·9-s + 0.288·12-s − 0.258·15-s − 3/4·16-s − 0.970·17-s − 0.223·20-s − 2/5·25-s + 0.192·27-s + 1/6·36-s − 0.657·37-s + 1.87·41-s + 1.21·43-s − 0.149·45-s − 0.433·48-s − 49-s − 0.560·51-s − 1.04·59-s − 0.129·60-s − 7/8·64-s − 0.977·67-s − 0.485·68-s − 0.230·75-s + 0.335·80-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6615 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6615 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.065470266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.065470266\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 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
−11.96674570528888352784590256031, −11.28078562151763228684236334049, −10.90551042382422400651026208050, −10.36016771654528399053265104347, −9.335539872920611783638396169880, −9.199866042579042052517471539219, −8.382212856146157064902519405280, −7.72009981120177449248701273285, −7.22495947379405190721708338248, −6.52575558803626067260081988600, −5.84891730358471501690019697703, −4.68155243915535530135391274133, −4.11619187207847055059962499072, −3.03094143106400761993417036804, −2.07915336094385051156894345324,
2.07915336094385051156894345324, 3.03094143106400761993417036804, 4.11619187207847055059962499072, 4.68155243915535530135391274133, 5.84891730358471501690019697703, 6.52575558803626067260081988600, 7.22495947379405190721708338248, 7.72009981120177449248701273285, 8.382212856146157064902519405280, 9.199866042579042052517471539219, 9.335539872920611783638396169880, 10.36016771654528399053265104347, 10.90551042382422400651026208050, 11.28078562151763228684236334049, 11.96674570528888352784590256031
