L(s) = 1 | − 5-s − 7-s − 3·11-s − 5.29·13-s − 5.29·17-s + 5.29·19-s + 5.29·23-s − 4·25-s + 6·29-s − 7·31-s + 35-s + 5.29·37-s − 5.29·41-s − 10.5·43-s − 6·49-s + 9·53-s + 3·55-s + 4·59-s + 5.29·65-s − 15.8·71-s − 3·73-s + 3·77-s + 4·79-s − 7·83-s + 5.29·85-s + 10.5·89-s + 5.29·91-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s − 0.904·11-s − 1.46·13-s − 1.28·17-s + 1.21·19-s + 1.10·23-s − 0.800·25-s + 1.11·29-s − 1.25·31-s + 0.169·35-s + 0.869·37-s − 0.826·41-s − 1.61·43-s − 0.857·49-s + 1.23·53-s + 0.404·55-s + 0.520·59-s + 0.656·65-s − 1.88·71-s − 0.351·73-s + 0.341·77-s + 0.450·79-s − 0.768·83-s + 0.573·85-s + 1.12·89-s + 0.554·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8592993113\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8592993113\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 5.29T + 13T^{2} \) |
| 17 | \( 1 + 5.29T + 17T^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 23 | \( 1 - 5.29T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 - 5.29T + 37T^{2} \) |
| 41 | \( 1 + 5.29T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 + 3T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 7T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−7.83004878135850021260322533109, −7.24733687005638983033153791930, −6.79285986630475731406598553040, −5.76677220661700548428553595677, −4.98317179351683571623961811249, −4.57400648671891606336547090280, −3.42529747916475792423809874207, −2.80590209330872918880931554656, −1.94430731856748231700839014441, −0.44747664183456919828886793529,
0.44747664183456919828886793529, 1.94430731856748231700839014441, 2.80590209330872918880931554656, 3.42529747916475792423809874207, 4.57400648671891606336547090280, 4.98317179351683571623961811249, 5.76677220661700548428553595677, 6.79285986630475731406598553040, 7.24733687005638983033153791930, 7.83004878135850021260322533109