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
−9.523366877292422814984696329564, −8.697247290866287187477316364822, −7.85289516046289477495230842491, −7.11658178793876040650218536499, −6.19329046921102766141414708427, −5.88069032588630055925593379149, −4.40855404771172149473939239722, −3.24105441868342084118860785511, −2.18013348846041940590421609389, −0.928731717517437252013406879432,
0.000082522817669903067956342846, 2.54544096326984073703104402784, 3.45795694770782677015210768692, 3.81850526076762218552135036969, 5.10056151211076923541789858835, 6.13990248664045716787253360841, 6.65890073317884287521046556548, 7.71135001410847517030936148505, 9.024852076917501812353289373354, 9.535186955014494163673057551494