L(s) = 1 | + (−1.31 − 1.48i)3-s + (0.354 + 0.935i)4-s + (1.31 + 0.159i)5-s + (−0.354 + 2.92i)9-s + (0.885 + 0.464i)11-s + (0.922 − 1.75i)12-s + (−1.49 − 2.16i)15-s + (−0.748 + 0.663i)16-s + (0.317 + 1.28i)20-s − 0.929i·23-s + (0.736 + 0.181i)25-s + (3.17 − 2.19i)27-s + (−0.869 − 1.65i)31-s + (−0.475 − 1.92i)33-s + (−2.85 + 0.704i)36-s + (0.180 − 0.159i)37-s + ⋯ |
L(s) = 1 | + (−1.31 − 1.48i)3-s + (0.354 + 0.935i)4-s + (1.31 + 0.159i)5-s + (−0.354 + 2.92i)9-s + (0.885 + 0.464i)11-s + (0.922 − 1.75i)12-s + (−1.49 − 2.16i)15-s + (−0.748 + 0.663i)16-s + (0.317 + 1.28i)20-s − 0.929i·23-s + (0.736 + 0.181i)25-s + (3.17 − 2.19i)27-s + (−0.869 − 1.65i)31-s + (−0.475 − 1.92i)33-s + (−2.85 + 0.704i)36-s + (0.180 − 0.159i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8170986384\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8170986384\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.885 - 0.464i)T \) |
| 53 | \( 1 + (0.970 - 0.239i)T \) |
good | 2 | \( 1 + (-0.354 - 0.935i)T^{2} \) |
| 3 | \( 1 + (1.31 + 1.48i)T + (-0.120 + 0.992i)T^{2} \) |
| 5 | \( 1 + (-1.31 - 0.159i)T + (0.970 + 0.239i)T^{2} \) |
| 7 | \( 1 + (0.354 + 0.935i)T^{2} \) |
| 13 | \( 1 + (0.748 + 0.663i)T^{2} \) |
| 17 | \( 1 + (-0.885 + 0.464i)T^{2} \) |
| 19 | \( 1 + (-0.748 - 0.663i)T^{2} \) |
| 23 | \( 1 + 0.929iT - T^{2} \) |
| 29 | \( 1 + (-0.568 + 0.822i)T^{2} \) |
| 31 | \( 1 + (0.869 + 1.65i)T + (-0.568 + 0.822i)T^{2} \) |
| 37 | \( 1 + (-0.180 + 0.159i)T + (0.120 - 0.992i)T^{2} \) |
| 41 | \( 1 + (0.568 + 0.822i)T^{2} \) |
| 43 | \( 1 + (-0.120 - 0.992i)T^{2} \) |
| 47 | \( 1 + (-0.180 - 1.48i)T + (-0.970 + 0.239i)T^{2} \) |
| 59 | \( 1 + (-0.0854 - 0.704i)T + (-0.970 + 0.239i)T^{2} \) |
| 61 | \( 1 + (0.885 + 0.464i)T^{2} \) |
| 67 | \( 1 + (1.53 - 0.583i)T + (0.748 - 0.663i)T^{2} \) |
| 71 | \( 1 + (0.317 - 0.358i)T + (-0.120 - 0.992i)T^{2} \) |
| 73 | \( 1 + (0.885 - 0.464i)T^{2} \) |
| 79 | \( 1 + (-0.354 + 0.935i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-1.71 + 0.423i)T + (0.885 - 0.464i)T^{2} \) |
| 97 | \( 1 + (-0.213 + 1.75i)T + (-0.970 - 0.239i)T^{2} \) |
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
−11.17492182555859363020735243039, −10.25675512288739506749312315433, −9.051424934152051261670496321501, −7.84893482960921704964095383772, −7.11651865403696387465789054052, −6.33341809188633799411340066841, −5.85294682929509876502362498055, −4.51993018342766279277972189439, −2.49868359450360690008596218646, −1.66263010630177631703748991441,
1.41835959093538318354190762545, 3.45885758688570570918995039298, 4.82406045191106846611777907437, 5.46503086845863140338043556915, 6.12311743112634394362730765286, 6.74926372711013070217222591917, 9.105483378952283688471285502119, 9.371288451423661414063899336511, 10.22515946494790317940424541619, 10.74820512322798699434435980650