| L(s) = 1 | − 3·5-s + 2·9-s − 6·13-s + 4·17-s + 2·25-s − 4·29-s + 16·37-s − 6·45-s − 2·49-s − 4·53-s − 12·61-s + 18·65-s + 16·73-s − 5·81-s − 12·85-s + 8·89-s + 12·101-s + 12·113-s − 12·117-s + 14·121-s + 10·125-s + 127-s + 131-s + 137-s + 139-s + 12·145-s + 149-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 2/3·9-s − 1.66·13-s + 0.970·17-s + 2/5·25-s − 0.742·29-s + 2.63·37-s − 0.894·45-s − 2/7·49-s − 0.549·53-s − 1.53·61-s + 2.23·65-s + 1.87·73-s − 5/9·81-s − 1.30·85-s + 0.847·89-s + 1.19·101-s + 1.12·113-s − 1.10·117-s + 1.27·121-s + 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.996·145-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216320 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216320 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.054021929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.054021929\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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
−9.161406970312219303621414977573, −8.426704341278964553409018236008, −7.80088345700239648051104421439, −7.63337390192865185684141254904, −7.46027327461613635973650249135, −6.75125194809500826014735561856, −6.16997170305890805734665316051, −5.57686728851440787487480708796, −4.85306557582909413470246785364, −4.52957347063305386105844861155, −3.98982166312148247461350686235, −3.35902937267377933318283659238, −2.73752717014157910656360974870, −1.87254478553143419238258150274, −0.64269411723060939117885709818,
0.64269411723060939117885709818, 1.87254478553143419238258150274, 2.73752717014157910656360974870, 3.35902937267377933318283659238, 3.98982166312148247461350686235, 4.52957347063305386105844861155, 4.85306557582909413470246785364, 5.57686728851440787487480708796, 6.16997170305890805734665316051, 6.75125194809500826014735561856, 7.46027327461613635973650249135, 7.63337390192865185684141254904, 7.80088345700239648051104421439, 8.426704341278964553409018236008, 9.161406970312219303621414977573
