L(s) = 1 | + i·2-s − 4-s − i·7-s − i·8-s + 3·9-s + 4·11-s + 6i·13-s + 14-s + 16-s + 2i·17-s + 3i·18-s + 4i·22-s − 6·26-s + i·28-s − 6·29-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.377i·7-s − 0.353i·8-s + 9-s + 1.20·11-s + 1.66i·13-s + 0.267·14-s + 0.250·16-s + 0.485i·17-s + 0.707i·18-s + 0.852i·22-s − 1.17·26-s + 0.188i·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.19252 + 0.737021i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.19252 + 0.737021i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 - 3T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−11.73524507409310901629681611586, −10.65002353250020221104707220904, −9.494747861583977445836362589452, −9.012486951053894611631767077035, −7.64306579165661627096830496648, −6.86420041488570947422359610279, −6.11189048839258655266713140206, −4.48247240717486246287022733395, −3.92999144609648339195991388200, −1.59233549458569435829285893892,
1.24175101525618634516311581904, 2.89477719381851478730103258526, 4.08160789816250886914506969336, 5.22914717908928257419633570309, 6.46439935707185610272543985490, 7.67144460496492996069726098658, 8.688116991019315822572852464589, 9.728098846078623639997935622348, 10.28837052092234625714461130138, 11.44513264493481635354231646084