L(s) = 1 | + (−0.484 − 1.32i)2-s + (−1.59 − 0.676i)3-s + (−1.53 + 1.28i)4-s + 1.67·5-s + (−0.126 + 2.44i)6-s − 2.28i·7-s + (2.45 + 1.41i)8-s + (2.08 + 2.15i)9-s + (−0.809 − 2.22i)10-s + 2.91i·11-s + (3.31 − 1.01i)12-s + 4.63i·13-s + (−3.03 + 1.10i)14-s + (−2.66 − 1.13i)15-s + (0.689 − 3.94i)16-s + 3.10i·17-s + ⋯ |
L(s) = 1 | + (−0.342 − 0.939i)2-s + (−0.920 − 0.390i)3-s + (−0.765 + 0.643i)4-s + 0.747·5-s + (−0.0516 + 0.998i)6-s − 0.864i·7-s + (0.866 + 0.499i)8-s + (0.695 + 0.718i)9-s + (−0.256 − 0.702i)10-s + 0.877i·11-s + (0.955 − 0.293i)12-s + 1.28i·13-s + (−0.812 + 0.295i)14-s + (−0.688 − 0.291i)15-s + (0.172 − 0.985i)16-s + 0.752i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.602 + 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.602 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.874168 - 0.435103i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.874168 - 0.435103i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.484 + 1.32i)T \) |
| 3 | \( 1 + (1.59 + 0.676i)T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 1.67T + 5T^{2} \) |
| 7 | \( 1 + 2.28iT - 7T^{2} \) |
| 11 | \( 1 - 2.91iT - 11T^{2} \) |
| 13 | \( 1 - 4.63iT - 13T^{2} \) |
| 17 | \( 1 - 3.10iT - 17T^{2} \) |
| 19 | \( 1 - 7.70T + 19T^{2} \) |
| 29 | \( 1 - 8.53T + 29T^{2} \) |
| 31 | \( 1 + 7.88iT - 31T^{2} \) |
| 37 | \( 1 + 3.97iT - 37T^{2} \) |
| 41 | \( 1 - 5.01iT - 41T^{2} \) |
| 43 | \( 1 - 7.62T + 43T^{2} \) |
| 47 | \( 1 + 4.61T + 47T^{2} \) |
| 53 | \( 1 - 0.427T + 53T^{2} \) |
| 59 | \( 1 - 9.10iT - 59T^{2} \) |
| 61 | \( 1 + 9.06iT - 61T^{2} \) |
| 67 | \( 1 + 8.29T + 67T^{2} \) |
| 71 | \( 1 - 4.33T + 71T^{2} \) |
| 73 | \( 1 - 6.91T + 73T^{2} \) |
| 79 | \( 1 + 6.46iT - 79T^{2} \) |
| 83 | \( 1 - 12.7iT - 83T^{2} \) |
| 89 | \( 1 + 4.52iT - 89T^{2} \) |
| 97 | \( 1 + 8.58T + 97T^{2} \) |
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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
−10.63546986921643140533271217685, −9.897499641934170867333860096147, −9.391765652099841010819089460081, −7.87207661208308946360316201577, −7.15301690347573097472472347947, −6.06654418197863261361220091671, −4.82339519610856135285346526289, −4.01554857820772419319339299247, −2.19428478355843384188498935376, −1.16181856835889047404984239248,
0.931535538243228739983574378738, 3.17232213066464346939493469017, 4.94657479717583705165169478668, 5.54079307434036932823878270924, 6.08197273586085588710277126640, 7.16554832347776015131736645796, 8.296165600779959384550105999164, 9.221168385570490416423849901064, 9.943010158877751834895088863145, 10.61581058423505507994988298388