| L(s) = 1 | − 3-s − 4-s + 4·5-s + 9-s + 12-s − 4·15-s + 16-s − 4·20-s + 11·25-s − 27-s − 36-s + 4·45-s + 16·47-s − 48-s − 7·49-s + 4·60-s − 64-s − 11·75-s + 16·79-s + 4·80-s + 81-s + 24·83-s − 11·100-s + 108-s − 28·109-s − 6·121-s + 24·125-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s + 1.78·5-s + 1/3·9-s + 0.288·12-s − 1.03·15-s + 1/4·16-s − 0.894·20-s + 11/5·25-s − 0.192·27-s − 1/6·36-s + 0.596·45-s + 2.33·47-s − 0.144·48-s − 49-s + 0.516·60-s − 1/8·64-s − 1.27·75-s + 1.80·79-s + 0.447·80-s + 1/9·81-s + 2.63·83-s − 1.09·100-s + 0.0962·108-s − 2.68·109-s − 0.545·121-s + 2.14·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132300 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132300 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.659397097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.659397097\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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
−9.382079667207611550181902865879, −9.166320354255721860880853713173, −8.527606124134460565465988984196, −7.927461294375509403356736877594, −7.36649750923666807230761191162, −6.60155933017990789780037515756, −6.42411140065541329502446951920, −5.74295706139379495360936903101, −5.38369296605186321956951583532, −4.93174302448540981311704327466, −4.27573257294083501838383539918, −3.50341666432398320311312850768, −2.59942620037353270460340022776, −1.94560952433490753418597398271, −0.988344135984823100739915057559,
0.988344135984823100739915057559, 1.94560952433490753418597398271, 2.59942620037353270460340022776, 3.50341666432398320311312850768, 4.27573257294083501838383539918, 4.93174302448540981311704327466, 5.38369296605186321956951583532, 5.74295706139379495360936903101, 6.42411140065541329502446951920, 6.60155933017990789780037515756, 7.36649750923666807230761191162, 7.927461294375509403356736877594, 8.527606124134460565465988984196, 9.166320354255721860880853713173, 9.382079667207611550181902865879
