L(s) = 1 | + 2.23i·3-s − 2.23i·5-s + (2.12 + 1.58i)7-s − 2.00·9-s + (1 − 3.16i)11-s − 2.82·13-s + 5.00·15-s + 5.65·17-s − 1.41·19-s + (−3.53 + 4.74i)21-s + 5·23-s + 2.23i·27-s − 9.48i·29-s + 2.23i·31-s + (7.07 + 2.23i)33-s + ⋯ |
L(s) = 1 | + 1.29i·3-s − 0.999i·5-s + (0.801 + 0.597i)7-s − 0.666·9-s + (0.301 − 0.953i)11-s − 0.784·13-s + 1.29·15-s + 1.37·17-s − 0.324·19-s + (−0.771 + 1.03i)21-s + 1.04·23-s + 0.430i·27-s − 1.76i·29-s + 0.401i·31-s + (1.23 + 0.389i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.898372422\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.898372422\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.12 - 1.58i)T \) |
| 11 | \( 1 + (-1 + 3.16i)T \) |
good | 3 | \( 1 - 2.23iT - 3T^{2} \) |
| 5 | \( 1 + 2.23iT - 5T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 + 1.41T + 19T^{2} \) |
| 23 | \( 1 - 5T + 23T^{2} \) |
| 29 | \( 1 + 9.48iT - 29T^{2} \) |
| 31 | \( 1 - 2.23iT - 31T^{2} \) |
| 37 | \( 1 - 11T + 37T^{2} \) |
| 41 | \( 1 + 9.89T + 41T^{2} \) |
| 43 | \( 1 - 3.16iT - 43T^{2} \) |
| 47 | \( 1 - 4.47iT - 47T^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 + 2.23iT - 59T^{2} \) |
| 61 | \( 1 - 4.24T + 61T^{2} \) |
| 67 | \( 1 + 9T + 67T^{2} \) |
| 71 | \( 1 - T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 + 6.32iT - 79T^{2} \) |
| 83 | \( 1 + 1.41T + 83T^{2} \) |
| 89 | \( 1 - 6.70iT - 89T^{2} \) |
| 97 | \( 1 - 15.6iT - 97T^{2} \) |
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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
−9.677149143221442618673129843023, −9.079558622409624629237025124061, −8.387012418980829890106661727943, −7.66027813533738121813874938738, −6.13120832594302514319763091226, −5.22984579948193774283988694324, −4.80593677984837704029640291901, −3.88124427005840656420863255054, −2.73902885068759476310571843518, −1.08595405206762833264161296480,
1.13735646189470533455780803352, 2.13235024101576415339644440944, 3.20808473773921154589710632675, 4.51792045636363850794880734999, 5.51609745643045463276098271538, 6.81371536135315739384037676534, 7.10934025229540875007703734917, 7.64338505851487617278165625535, 8.568444267229292225752892120641, 9.829978510888203689563688006426