| L(s) = 1 | − 2·2-s + 2·4-s − 5-s + 2·10-s + 4·13-s − 4·16-s − 2·20-s + 25-s − 8·26-s + 4·31-s + 8·32-s + 12·37-s + 4·41-s + 8·43-s − 2·49-s − 2·50-s + 8·52-s + 12·53-s − 8·62-s − 8·64-s − 4·65-s + 8·67-s + 8·71-s − 24·74-s − 20·79-s + 4·80-s − 8·82-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.447·5-s + 0.632·10-s + 1.10·13-s − 16-s − 0.447·20-s + 1/5·25-s − 1.56·26-s + 0.718·31-s + 1.41·32-s + 1.97·37-s + 0.624·41-s + 1.21·43-s − 2/7·49-s − 0.282·50-s + 1.10·52-s + 1.64·53-s − 1.01·62-s − 64-s − 0.496·65-s + 0.977·67-s + 0.949·71-s − 2.78·74-s − 2.25·79-s + 0.447·80-s − 0.883·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9350613619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9350613619\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 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
−8.365915193538754979177037091557, −8.037459296041924420717600339809, −7.62162418442154201838606402336, −7.23471517259649741126879041265, −6.69912633354319310059981584084, −6.21921539586360103344385594375, −5.84751035341134690973814753725, −5.16279770957833569258475475144, −4.46443648030601947741578978355, −4.12272106277996921302163179225, −3.55056711435809800825019349720, −2.69319837946002029227756456728, −2.24365211718443199428950526279, −1.21594982011144473293051325449, −0.72180896889441779745148652277,
0.72180896889441779745148652277, 1.21594982011144473293051325449, 2.24365211718443199428950526279, 2.69319837946002029227756456728, 3.55056711435809800825019349720, 4.12272106277996921302163179225, 4.46443648030601947741578978355, 5.16279770957833569258475475144, 5.84751035341134690973814753725, 6.21921539586360103344385594375, 6.69912633354319310059981584084, 7.23471517259649741126879041265, 7.62162418442154201838606402336, 8.037459296041924420717600339809, 8.365915193538754979177037091557
