| L(s) = 1 | − 13.8i·7-s − 22i·13-s + 27.7·19-s + 25·25-s − 41.5i·31-s + 26i·37-s + 83.1·43-s − 142.·49-s − 74i·61-s − 55.4·67-s − 46·73-s + 69.2i·79-s − 304.·91-s − 2·97-s + 69.2i·103-s + ⋯ |
| L(s) = 1 | − 1.97i·7-s − 1.69i·13-s + 1.45·19-s + 25-s − 1.34i·31-s + 0.702i·37-s + 1.93·43-s − 2.91·49-s − 1.21i·61-s − 0.827·67-s − 0.630·73-s + 0.876i·79-s − 3.34·91-s − 0.0206·97-s + 0.672i·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.943527035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.943527035\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 25T^{2} \) |
| 7 | \( 1 + 13.8iT - 49T^{2} \) |
| 11 | \( 1 + 121T^{2} \) |
| 13 | \( 1 + 22iT - 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 - 27.7T + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 41.5iT - 961T^{2} \) |
| 37 | \( 1 - 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 1.68e3T^{2} \) |
| 43 | \( 1 - 83.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 + 3.48e3T^{2} \) |
| 61 | \( 1 + 74iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 55.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 46T + 5.32e3T^{2} \) |
| 79 | \( 1 - 69.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 + 7.92e3T^{2} \) |
| 97 | \( 1 + 2T + 9.40e3T^{2} \) |
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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
−8.318786994785681104187533261401, −7.57891287855260520130377912164, −7.29512663534570536398711762262, −6.23248525516154022199607206244, −5.31212491811102545489852717806, −4.46584739313504489436522546372, −3.58586603657582804014793119464, −2.85985099796296523317651667628, −1.14867004151208707733286682957, −0.54124297949325164943205189098,
1.40893859266502846699019968045, 2.40006293295397425741309633681, 3.16570136011276093075230831840, 4.42856291239911575569499403689, 5.26077982744396780234714156184, 5.90605606682201588505037692799, 6.75089411084164583343880711237, 7.53540328103783595411652544787, 8.667574432823910434364601612008, 9.062916890448988273190678531347
