L(s) = 1 | − 0.585i·5-s + 0.828i·7-s − 2.82·11-s − 2.82·13-s + 2.58i·17-s + 5.65i·19-s − 6.82·23-s + 4.65·25-s − 3.41i·29-s + 8.82i·31-s + 0.485·35-s − 7.65·37-s + 5.41i·41-s + 1.65i·43-s + 4.48·47-s + ⋯ |
L(s) = 1 | − 0.261i·5-s + 0.313i·7-s − 0.852·11-s − 0.784·13-s + 0.627i·17-s + 1.29i·19-s − 1.42·23-s + 0.931·25-s − 0.634i·29-s + 1.58i·31-s + 0.0820·35-s − 1.25·37-s + 0.845i·41-s + 0.252i·43-s + 0.654·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7159483209\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7159483209\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 0.585iT - 5T^{2} \) |
| 7 | \( 1 - 0.828iT - 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 - 2.58iT - 17T^{2} \) |
| 19 | \( 1 - 5.65iT - 19T^{2} \) |
| 23 | \( 1 + 6.82T + 23T^{2} \) |
| 29 | \( 1 + 3.41iT - 29T^{2} \) |
| 31 | \( 1 - 8.82iT - 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 - 5.41iT - 41T^{2} \) |
| 43 | \( 1 - 1.65iT - 43T^{2} \) |
| 47 | \( 1 - 4.48T + 47T^{2} \) |
| 53 | \( 1 + 9.07iT - 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 + 3.65T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 10.4iT - 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 3.75iT - 89T^{2} \) |
| 97 | \( 1 - 2.34T + 97T^{2} \) |
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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
−10.29317110179427784887164740436, −9.257192126274612856138578594839, −8.320499599362063731320412375970, −7.81740640802887611826150140983, −6.73760494218947621075126231340, −5.76347387890423002773379728221, −5.04339367096477744846169214714, −3.99081539792909957437747345814, −2.82430964362749475080910665349, −1.67208705540559493262913110002,
0.29286805446619933560333623045, 2.19638920872998885645758637939, 3.08367683441855913665075854293, 4.37996645894102702388425627831, 5.15242546716902508036397897380, 6.15686073907591485825550347302, 7.26824704243720903316685394048, 7.59459710264408908853655607753, 8.796675640253452903907022083541, 9.499268722837190812756418712633