L(s) = 1 | + (−0.471 − 0.881i)2-s + (−0.555 + 0.831i)4-s + (0.831 − 0.555i)5-s + (1.83 + 0.761i)7-s + (0.995 + 0.0980i)8-s + (−0.923 + 0.382i)9-s + (−0.881 − 0.471i)10-s + (0.980 + 0.195i)11-s + (0.482 + 0.322i)13-s + (−0.195 − 1.98i)14-s + (−0.382 − 0.923i)16-s + (−0.897 + 0.897i)17-s + (0.773 + 0.634i)18-s + i·20-s + (−0.290 − 0.956i)22-s + ⋯ |
L(s) = 1 | + (−0.471 − 0.881i)2-s + (−0.555 + 0.831i)4-s + (0.831 − 0.555i)5-s + (1.83 + 0.761i)7-s + (0.995 + 0.0980i)8-s + (−0.923 + 0.382i)9-s + (−0.881 − 0.471i)10-s + (0.980 + 0.195i)11-s + (0.482 + 0.322i)13-s + (−0.195 − 1.98i)14-s + (−0.382 − 0.923i)16-s + (−0.897 + 0.897i)17-s + (0.773 + 0.634i)18-s + i·20-s + (−0.290 − 0.956i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.365719218\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.365719218\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.471 + 0.881i)T \) |
| 5 | \( 1 + (-0.831 + 0.555i)T \) |
| 11 | \( 1 + (-0.980 - 0.195i)T \) |
good | 3 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 7 | \( 1 + (-1.83 - 0.761i)T + (0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + (-0.482 - 0.322i)T + (0.382 + 0.923i)T^{2} \) |
| 17 | \( 1 + (0.897 - 0.897i)T - iT^{2} \) |
| 19 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 23 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 31 | \( 1 + 1.96iT - T^{2} \) |
| 37 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 41 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 43 | \( 1 + (0.344 - 1.72i)T + (-0.923 - 0.382i)T^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 59 | \( 1 + (1.38 - 0.923i)T + (0.382 - 0.923i)T^{2} \) |
| 61 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 67 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 71 | \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 73 | \( 1 + (0.871 - 0.360i)T + (0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 - iT^{2} \) |
| 83 | \( 1 + (0.322 - 0.482i)T + (-0.382 - 0.923i)T^{2} \) |
| 89 | \( 1 + (-0.425 + 1.02i)T + (-0.707 - 0.707i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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
−8.892281124466112631488000183966, −8.255446977684485836862223127372, −7.66207017966694143145151818505, −6.22801333790969524241928518759, −5.65298152779149544711191150293, −4.57432996452974775120744090910, −4.30743779237906325060057353006, −2.75142996869642749304531677468, −1.89821120383192423275632145195, −1.45766682760243129413332106012,
1.11093024978031131171729688287, 1.96872960749537090044614560107, 3.39240488938107693325450767183, 4.50980797917004017658775576286, 5.18937859100794343839062360052, 5.88434974756939255096402448989, 6.76840918577435636245774483739, 7.17590767310423170246527197389, 8.140509142809544914455671195157, 8.826087663677672526484568187323