L(s) = 1 | − 1.53·2-s + 1.34i·3-s + 0.347·4-s + 3.53i·5-s − 2.06i·6-s + 0.347i·7-s + 2.53·8-s + 1.18·9-s − 5.41i·10-s + 1.75i·11-s + 0.467i·12-s − 3.29·13-s − 0.532i·14-s − 4.75·15-s − 4.57·16-s + ⋯ |
L(s) = 1 | − 1.08·2-s + 0.777i·3-s + 0.173·4-s + 1.57i·5-s − 0.842i·6-s + 0.131i·7-s + 0.895·8-s + 0.394·9-s − 1.71i·10-s + 0.530i·11-s + 0.135i·12-s − 0.912·13-s − 0.142i·14-s − 1.22·15-s − 1.14·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 - 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.120476 + 0.560358i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.120476 + 0.560358i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 \) |
good | 2 | \( 1 + 1.53T + 2T^{2} \) |
| 3 | \( 1 - 1.34iT - 3T^{2} \) |
| 5 | \( 1 - 3.53iT - 5T^{2} \) |
| 7 | \( 1 - 0.347iT - 7T^{2} \) |
| 11 | \( 1 - 1.75iT - 11T^{2} \) |
| 13 | \( 1 + 3.29T + 13T^{2} \) |
| 19 | \( 1 + 1.53T + 19T^{2} \) |
| 23 | \( 1 + 2.81iT - 23T^{2} \) |
| 29 | \( 1 + 1.18iT - 29T^{2} \) |
| 31 | \( 1 - 7.10iT - 31T^{2} \) |
| 37 | \( 1 + 3.92iT - 37T^{2} \) |
| 41 | \( 1 + 4.92iT - 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 5.12T + 47T^{2} \) |
| 53 | \( 1 - 8.36T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 4.41iT - 61T^{2} \) |
| 67 | \( 1 - 8.07T + 67T^{2} \) |
| 71 | \( 1 - 11.2iT - 71T^{2} \) |
| 73 | \( 1 - 1.70iT - 73T^{2} \) |
| 79 | \( 1 + 11.5iT - 79T^{2} \) |
| 83 | \( 1 - 2.14T + 83T^{2} \) |
| 89 | \( 1 - 6.41T + 89T^{2} \) |
| 97 | \( 1 - 4.08iT - 97T^{2} \) |
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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
−11.84255026860971441132558350344, −10.66005984176071395569112770724, −10.28990565882236652392885284002, −9.662253700239910545347452963208, −8.584883553984798683206674406707, −7.31399589565976057666917371243, −6.82406349876047458226338411314, −5.02534235640028237099066424120, −3.82514743375886389301306111165, −2.28102035559970379373319182444,
0.63251848739655294396149255692, 1.80642340625490447407844020161, 4.27344633400243721553007143700, 5.28284287651621414997897738986, 6.83656809628690498277025332840, 7.904655181951897699703205250418, 8.408675709118140865423695966428, 9.445568274563134837174397657717, 10.04606212210840091644096849879, 11.46276440062832229473703189768