L(s) = 1 | + 2-s + (−0.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.866 + 0.5i)11-s + (−0.5 + 0.866i)12-s + (−0.866 + 0.5i)13-s + (−0.5 + 0.866i)14-s + 16-s + (0.866 + 0.5i)17-s + (−0.866 − 0.5i)19-s + (−0.499 − 0.866i)21-s + (−0.866 + 0.5i)22-s + ⋯ |
L(s) = 1 | + 2-s + (−0.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.866 + 0.5i)11-s + (−0.5 + 0.866i)12-s + (−0.866 + 0.5i)13-s + (−0.5 + 0.866i)14-s + 16-s + (0.866 + 0.5i)17-s + (−0.866 − 0.5i)19-s + (−0.499 − 0.866i)21-s + (−0.866 + 0.5i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.658199542\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.658199542\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 31 | \( 1 + iT \) |
good | 3 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.502070614317316322226420419406, −8.226156267350649778800977662897, −7.52989204226947053415516716560, −6.60722752919358680626330544788, −5.85379846085590710231635614065, −5.19370257786938874377602238148, −4.62520744386184460652465717094, −3.85247093224444412905809693895, −2.72979899478129858395223708089, −2.07075257243076135103761956433,
0.72956131242624455038558864009, 2.04214823574157853483346036175, 3.11392807792812071065267307891, 3.80949000323466071605269263545, 4.96441171921821926305060403306, 5.52886430571236832410393137298, 6.44314674802980320164966725556, 6.88784323145163958842010810664, 7.68088732411257370480406358453, 8.116218655223335957277832839653