L(s) = 1 | + 1.28·3-s − 5-s − 0.791·7-s − 1.35·9-s + 5.58·11-s + 0.759·13-s − 1.28·15-s − 7.02·17-s + 4.40·19-s − 1.01·21-s + 3.48·23-s + 25-s − 5.58·27-s − 4.93·29-s − 5.65·31-s + 7.16·33-s + 0.791·35-s + 2.56·37-s + 0.973·39-s − 1.22·41-s − 6.23·43-s + 1.35·45-s − 6.02·47-s − 6.37·49-s − 9.00·51-s − 8.35·53-s − 5.58·55-s + ⋯ |
L(s) = 1 | + 0.740·3-s − 0.447·5-s − 0.299·7-s − 0.452·9-s + 1.68·11-s + 0.210·13-s − 0.330·15-s − 1.70·17-s + 1.01·19-s − 0.221·21-s + 0.727·23-s + 0.200·25-s − 1.07·27-s − 0.915·29-s − 1.01·31-s + 1.24·33-s + 0.133·35-s + 0.421·37-s + 0.155·39-s − 0.191·41-s − 0.951·43-s + 0.202·45-s − 0.878·47-s − 0.910·49-s − 1.26·51-s − 1.14·53-s − 0.753·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 - 1.28T + 3T^{2} \) |
| 7 | \( 1 + 0.791T + 7T^{2} \) |
| 11 | \( 1 - 5.58T + 11T^{2} \) |
| 13 | \( 1 - 0.759T + 13T^{2} \) |
| 17 | \( 1 + 7.02T + 17T^{2} \) |
| 19 | \( 1 - 4.40T + 19T^{2} \) |
| 23 | \( 1 - 3.48T + 23T^{2} \) |
| 29 | \( 1 + 4.93T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 - 2.56T + 37T^{2} \) |
| 41 | \( 1 + 1.22T + 41T^{2} \) |
| 43 | \( 1 + 6.23T + 43T^{2} \) |
| 47 | \( 1 + 6.02T + 47T^{2} \) |
| 53 | \( 1 + 8.35T + 53T^{2} \) |
| 59 | \( 1 + 9.38T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 6.07T + 67T^{2} \) |
| 71 | \( 1 - 3.00T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 + 8.59T + 79T^{2} \) |
| 83 | \( 1 - 6.43T + 83T^{2} \) |
| 89 | \( 1 + 18.5T + 89T^{2} \) |
| 97 | \( 1 - 2.63T + 97T^{2} \) |
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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
−7.51045131793989146965755072294, −6.75665200941712133989846717167, −6.36464810650629533202215978785, −5.34596456210836385177626331301, −4.52139320450597536553218189513, −3.61783090159664816789394259696, −3.36540791097433503519797949028, −2.26441017741009549029996490877, −1.38444681704797425284282462107, 0,
1.38444681704797425284282462107, 2.26441017741009549029996490877, 3.36540791097433503519797949028, 3.61783090159664816789394259696, 4.52139320450597536553218189513, 5.34596456210836385177626331301, 6.36464810650629533202215978785, 6.75665200941712133989846717167, 7.51045131793989146965755072294