L(s) = 1 | + 3-s − 3.56i·5-s − i·7-s + 9-s + 5.56i·11-s + (3.56 + 0.561i)13-s − 3.56i·15-s − 6.68·17-s − 1.56i·19-s − i·21-s + 6.68·23-s − 7.68·25-s + 27-s + 1.56·29-s − 6.24i·31-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.59i·5-s − 0.377i·7-s + 0.333·9-s + 1.67i·11-s + (0.987 + 0.155i)13-s − 0.919i·15-s − 1.62·17-s − 0.358i·19-s − 0.218i·21-s + 1.39·23-s − 1.53·25-s + 0.192·27-s + 0.289·29-s − 1.12i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.226208065\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.226208065\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (-3.56 - 0.561i)T \) |
good | 5 | \( 1 + 3.56iT - 5T^{2} \) |
| 11 | \( 1 - 5.56iT - 11T^{2} \) |
| 17 | \( 1 + 6.68T + 17T^{2} \) |
| 19 | \( 1 + 1.56iT - 19T^{2} \) |
| 23 | \( 1 - 6.68T + 23T^{2} \) |
| 29 | \( 1 - 1.56T + 29T^{2} \) |
| 31 | \( 1 + 6.24iT - 31T^{2} \) |
| 37 | \( 1 + 10.6iT - 37T^{2} \) |
| 41 | \( 1 + 4iT - 41T^{2} \) |
| 43 | \( 1 - 6.43T + 43T^{2} \) |
| 47 | \( 1 + 10.2iT - 47T^{2} \) |
| 53 | \( 1 - 4.87T + 53T^{2} \) |
| 59 | \( 1 - 4.24iT - 59T^{2} \) |
| 61 | \( 1 + 1.56T + 61T^{2} \) |
| 67 | \( 1 + 1.12iT - 67T^{2} \) |
| 71 | \( 1 - 9.36iT - 71T^{2} \) |
| 73 | \( 1 + 11.5iT - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 - 8iT - 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.404438760615022624750477585068, −7.34569256559053014845066431372, −7.02379786265701008616840023410, −5.88015115712191209560740827840, −4.94700145420831985294500508616, −4.34862841670481128355950434843, −3.92600550681014499371874721798, −2.40807401578216843026368995082, −1.69891793740178976989006057632, −0.60195638212821289629729008645,
1.24835618354111413881828906580, 2.62985303744714780380719673441, 3.04028163177262854340587197649, 3.67896006655227491468181980439, 4.77314176298719275962472585228, 5.96884309363708772361357289364, 6.39871997780398983118649022553, 6.99230336159348698331274961895, 7.920187737871581072631889931547, 8.687233120949223106878825266878