L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + 5i·7-s + i·8-s − 9-s − 11-s − i·12-s + 2i·13-s + 5·14-s + 16-s + i·17-s + i·18-s + 7·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 1.88i·7-s + 0.353i·8-s − 0.333·9-s − 0.301·11-s − 0.288i·12-s + 0.554i·13-s + 1.33·14-s + 0.250·16-s + 0.242i·17-s + 0.235i·18-s + 1.60·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.227981557\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.227981557\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 - iT \) |
good | 7 | \( 1 - 5iT - 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 - 3iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 9iT - 43T^{2} \) |
| 47 | \( 1 + 7iT - 47T^{2} \) |
| 53 | \( 1 - 11iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 13iT - 67T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 3T + 79T^{2} \) |
| 83 | \( 1 + 14iT - 83T^{2} \) |
| 89 | \( 1 + 4T + 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−9.151068943723225360059724254827, −8.676635831766271897420256231594, −7.975628870477223772557022726042, −6.70511439410044147750511525501, −5.74990505732547804528468477413, −5.18291280135879573308046411090, −4.42717894193652637504258172345, −3.14482496881194778708854965669, −2.70291583363309397495499174844, −1.54279479822898448616219573379,
0.43509260223608025489311448249, 1.36590696849494497851558601108, 3.07641607559140570256451621163, 3.81676111435583416335504296087, 4.89070280874311717662520231610, 5.51988700155461781919982185143, 6.71030710522872342213364328218, 7.08060251782171112510088437948, 7.76955741657710922557876033221, 8.261827334836010057403941007254