L(s) = 1 | + (−0.721 − 0.692i)3-s + (0.822 − 0.568i)4-s + (−0.160 + 0.987i)7-s + (0.0402 + 0.999i)9-s + (−0.987 − 0.160i)12-s + (−0.866 + 0.5i)13-s + (0.354 − 0.935i)16-s + (−0.504 + 1.88i)19-s + (0.799 − 0.600i)21-s + (−0.316 + 0.948i)25-s + (0.663 − 0.748i)27-s + (0.428 + 0.903i)28-s + (0.300 − 0.198i)31-s + (0.600 + 0.799i)36-s + (0.119 + 1.97i)37-s + ⋯ |
L(s) = 1 | + (−0.721 − 0.692i)3-s + (0.822 − 0.568i)4-s + (−0.160 + 0.987i)7-s + (0.0402 + 0.999i)9-s + (−0.987 − 0.160i)12-s + (−0.866 + 0.5i)13-s + (0.354 − 0.935i)16-s + (−0.504 + 1.88i)19-s + (0.799 − 0.600i)21-s + (−0.316 + 0.948i)25-s + (0.663 − 0.748i)27-s + (0.428 + 0.903i)28-s + (0.300 − 0.198i)31-s + (0.600 + 0.799i)36-s + (0.119 + 1.97i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9578319647\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9578319647\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.721 + 0.692i)T \) |
| 7 | \( 1 + (0.160 - 0.987i)T \) |
| 13 | \( 1 + (0.866 - 0.5i)T \) |
good | 2 | \( 1 + (-0.822 + 0.568i)T^{2} \) |
| 5 | \( 1 + (0.316 - 0.948i)T^{2} \) |
| 11 | \( 1 + (-0.0804 + 0.996i)T^{2} \) |
| 17 | \( 1 + (0.970 - 0.239i)T^{2} \) |
| 19 | \( 1 + (0.504 - 1.88i)T + (-0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (0.996 - 0.0804i)T^{2} \) |
| 31 | \( 1 + (-0.300 + 0.198i)T + (0.391 - 0.919i)T^{2} \) |
| 37 | \( 1 + (-0.119 - 1.97i)T + (-0.992 + 0.120i)T^{2} \) |
| 41 | \( 1 + (0.534 + 0.845i)T^{2} \) |
| 43 | \( 1 + (-1.17 + 0.240i)T + (0.919 - 0.391i)T^{2} \) |
| 47 | \( 1 + (-0.774 - 0.632i)T^{2} \) |
| 53 | \( 1 + (-0.278 + 0.960i)T^{2} \) |
| 59 | \( 1 + (-0.663 - 0.748i)T^{2} \) |
| 61 | \( 1 + (-0.336 - 0.788i)T + (-0.692 + 0.721i)T^{2} \) |
| 67 | \( 1 + (-0.113 - 0.0405i)T + (0.774 + 0.632i)T^{2} \) |
| 71 | \( 1 + (0.999 - 0.0402i)T^{2} \) |
| 73 | \( 1 + (1.94 - 0.437i)T + (0.903 - 0.428i)T^{2} \) |
| 79 | \( 1 + (0.205 + 0.432i)T + (-0.632 + 0.774i)T^{2} \) |
| 83 | \( 1 + (-0.464 - 0.885i)T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 + (0.0520 - 0.515i)T + (-0.979 - 0.200i)T^{2} \) |
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
−8.735803051926116444967131653840, −7.88571273985254066975038290561, −7.25501167795639871196276012774, −6.46862183802766314343872829872, −5.89373000830750681446017969552, −5.41279217715684958874303532095, −4.45547205833839170596051115958, −3.01668120757810383059749222778, −2.11930224232846306341991006743, −1.45421331970783444125417464320,
0.59056737380499673850550789737, 2.32568037990883401648469757462, 3.17317025841011757224997506609, 4.14859953947148832532825946297, 4.66995323220170446381288331871, 5.72019615855751958062124905384, 6.51410627639396652381254345973, 7.13249669613202932115441578696, 7.67076240970671727452550530542, 8.715924838104211522158701464797