L(s) = 1 | − 1.41i·2-s + (0.724 − 1.57i)3-s − 2.00·4-s + (−2.22 − 1.02i)6-s + 2.82i·8-s + (−1.94 − 2.28i)9-s − 6.61i·11-s + (−1.44 + 3.14i)12-s + 4.00·16-s + 2.36i·17-s + (−3.22 + 2.75i)18-s − 8.34·19-s − 9.34·22-s + (4.44 + 2.04i)24-s + (−5.00 + 1.41i)27-s + ⋯ |
L(s) = 1 | − 0.999i·2-s + (0.418 − 0.908i)3-s − 1.00·4-s + (−0.908 − 0.418i)6-s + 1.00i·8-s + (−0.649 − 0.760i)9-s − 1.99i·11-s + (−0.418 + 0.908i)12-s + 1.00·16-s + 0.574i·17-s + (−0.760 + 0.649i)18-s − 1.91·19-s − 1.99·22-s + (0.908 + 0.418i)24-s + (−0.962 + 0.272i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 - 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.239286 + 1.09126i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.239286 + 1.09126i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + (-0.724 + 1.57i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 6.61iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 2.36iT - 17T^{2} \) |
| 19 | \( 1 + 8.34T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 0.460iT - 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 14.1iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 14.1iT - 83T^{2} \) |
| 89 | \( 1 + 12.7iT - 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−10.45277826903736670892428080480, −9.051711805165616786593532932683, −8.581294609966556397984440467977, −7.88321817747722108510718144503, −6.39494562166137588659466204729, −5.68472129577490533157520714386, −4.07720853240452779954542107476, −3.13996614076996600979206298663, −2.04832283639156490265104378621, −0.58663908602624407808599996061,
2.37672987597234901674568978512, 4.11122225182346979569168121830, 4.54193481752522462504601853870, 5.60323674393187027476246252548, 6.81609947955012021188071711995, 7.60135815847901278460459604886, 8.582494916036354261680445385096, 9.332454010540412247104919146260, 10.02963075738866796094413003815, 10.77134036012400299217004883952