| L(s) = 1 | + 3-s + 2·5-s + 9-s + 2·15-s + 17-s − 25-s + 27-s − 2·29-s + 8·31-s − 10·37-s + 6·41-s + 4·43-s + 2·45-s − 8·47-s − 7·49-s + 51-s − 2·53-s − 4·59-s − 10·61-s + 8·67-s − 12·71-s − 6·73-s − 75-s + 8·79-s + 81-s + 12·83-s + 2·85-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.516·15-s + 0.242·17-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s − 1.64·37-s + 0.937·41-s + 0.609·43-s + 0.298·45-s − 1.16·47-s − 49-s + 0.140·51-s − 0.274·53-s − 0.520·59-s − 1.28·61-s + 0.977·67-s − 1.42·71-s − 0.702·73-s − 0.115·75-s + 0.900·79-s + 1/9·81-s + 1.31·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.685877603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.685877603\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.50124042252695, −13.04514073332406, −12.58996916890862, −11.96735941827943, −11.66486856354396, −10.77094089664044, −10.58294184037399, −9.915436569278220, −9.565212373637639, −9.147408647896549, −8.618916610108707, −8.000534959402458, −7.691121570504353, −7.002421036043485, −6.418721496418418, −6.069157678988480, −5.438254765384222, −4.852874967607029, −4.385463285931381, −3.610472070337324, −3.115709002994329, −2.549439602254193, −1.840040899383038, −1.477115667269719, −0.5305978075746488,
0.5305978075746488, 1.477115667269719, 1.840040899383038, 2.549439602254193, 3.115709002994329, 3.610472070337324, 4.385463285931381, 4.852874967607029, 5.438254765384222, 6.069157678988480, 6.418721496418418, 7.002421036043485, 7.691121570504353, 8.000534959402458, 8.618916610108707, 9.147408647896549, 9.565212373637639, 9.915436569278220, 10.58294184037399, 10.77094089664044, 11.66486856354396, 11.96735941827943, 12.58996916890862, 13.04514073332406, 13.50124042252695
