| L(s) = 1 | + 4.47i·3-s + 5i·5-s − 13.4·7-s − 11.0·9-s − 22.3·15-s − 60.0i·21-s − 13.4·23-s − 25·25-s − 8.94i·27-s + 22i·29-s − 67.0i·35-s + 62·41-s − 40.2i·43-s − 55.0i·45-s − 93.9·47-s + ⋯ |
| L(s) = 1 | + 1.49i·3-s + i·5-s − 1.91·7-s − 1.22·9-s − 1.49·15-s − 2.85i·21-s − 0.583·23-s − 25-s − 0.331i·27-s + 0.758i·29-s − 1.91i·35-s + 1.51·41-s − 0.936i·43-s − 1.22i·45-s − 1.99·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0002360217117\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0002360217117\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5iT \) |
| good | 3 | \( 1 - 4.47iT - 9T^{2} \) |
| 7 | \( 1 + 13.4T + 49T^{2} \) |
| 11 | \( 1 + 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 + 361T^{2} \) |
| 23 | \( 1 + 13.4T + 529T^{2} \) |
| 29 | \( 1 - 22iT - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 - 62T + 1.68e3T^{2} \) |
| 43 | \( 1 + 40.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 93.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 + 3.48e3T^{2} \) |
| 61 | \( 1 + 58iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 67.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 147. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 142T + 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−9.535186955014494163673057551494, −9.024852076917501812353289373354, −7.71135001410847517030936148505, −6.65890073317884287521046556548, −6.13990248664045716787253360841, −5.10056151211076923541789858835, −3.81850526076762218552135036969, −3.45795694770782677015210768692, −2.54544096326984073703104402784, −0.000082522817669903067956342846,
0.928731717517437252013406879432, 2.18013348846041940590421609389, 3.24105441868342084118860785511, 4.40855404771172149473939239722, 5.88069032588630055925593379149, 6.19329046921102766141414708427, 7.11658178793876040650218536499, 7.85289516046289477495230842491, 8.697247290866287187477316364822, 9.523366877292422814984696329564
