L(s) = 1 | − 1.80·3-s + 4-s + 1.24·5-s + 2.24·9-s + 11-s − 1.80·12-s − 2.24·15-s + 16-s + 1.24·20-s − 0.445·23-s + 0.554·25-s − 2.24·27-s − 0.445·31-s − 1.80·33-s + 2.24·36-s − 1.80·37-s + 44-s + 2.80·45-s − 0.445·47-s − 1.80·48-s + 49-s + 1.24·53-s + 1.24·55-s − 1.80·59-s − 2.24·60-s + 64-s + 2·67-s + ⋯ |
L(s) = 1 | − 1.80·3-s + 4-s + 1.24·5-s + 2.24·9-s + 11-s − 1.80·12-s − 2.24·15-s + 16-s + 1.24·20-s − 0.445·23-s + 0.554·25-s − 2.24·27-s − 0.445·31-s − 1.80·33-s + 2.24·36-s − 1.80·37-s + 44-s + 2.80·45-s − 0.445·47-s − 1.80·48-s + 49-s + 1.24·53-s + 1.24·55-s − 1.80·59-s − 2.24·60-s + 64-s + 2·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.098934850\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.098934850\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - T^{2} \) |
| 3 | \( 1 + 1.80T + T^{2} \) |
| 5 | \( 1 - 1.24T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 0.445T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 0.445T + T^{2} \) |
| 37 | \( 1 + 1.80T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 0.445T + T^{2} \) |
| 53 | \( 1 - 1.24T + T^{2} \) |
| 59 | \( 1 + 1.80T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 2T + T^{2} \) |
| 71 | \( 1 + 1.80T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 0.445T + T^{2} \) |
| 97 | \( 1 - 1.24T + T^{2} \) |
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
−9.817156926755519163453905269555, −8.819714143810908806944350682872, −7.37200228352772388731079307821, −6.79640938010960423199608962464, −6.09943539099223884820044861384, −5.73449726855556323737518920849, −4.88218089659646155974992026746, −3.67127816214404962352341152175, −2.08108240462824727812101146251, −1.30460705206415413718097051199,
1.30460705206415413718097051199, 2.08108240462824727812101146251, 3.67127816214404962352341152175, 4.88218089659646155974992026746, 5.73449726855556323737518920849, 6.09943539099223884820044861384, 6.79640938010960423199608962464, 7.37200228352772388731079307821, 8.819714143810908806944350682872, 9.817156926755519163453905269555