| L(s) = 1 | − 5-s − 3.44·7-s − 2.44·11-s + 4.44·13-s + 5.44·17-s − 1.55·19-s + 23-s + 25-s + 1.89·29-s − 7·31-s + 3.44·35-s + 6.34·37-s + 7.89·41-s − 8.89·43-s − 2.44·47-s + 4.89·49-s − 4.34·53-s + 2.44·55-s − 1.89·59-s − 5.34·61-s − 4.44·65-s + 1.44·67-s + 3·71-s + 9.34·73-s + 8.44·77-s − 4·79-s − 10.3·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.30·7-s − 0.738·11-s + 1.23·13-s + 1.32·17-s − 0.355·19-s + 0.208·23-s + 0.200·25-s + 0.352·29-s − 1.25·31-s + 0.583·35-s + 1.04·37-s + 1.23·41-s − 1.35·43-s − 0.357·47-s + 0.699·49-s − 0.597·53-s + 0.330·55-s − 0.247·59-s − 0.684·61-s − 0.551·65-s + 0.177·67-s + 0.356·71-s + 1.09·73-s + 0.962·77-s − 0.450·79-s − 1.13·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 3.44T + 7T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 - 4.44T + 13T^{2} \) |
| 17 | \( 1 - 5.44T + 17T^{2} \) |
| 19 | \( 1 + 1.55T + 19T^{2} \) |
| 29 | \( 1 - 1.89T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 - 6.34T + 37T^{2} \) |
| 41 | \( 1 - 7.89T + 41T^{2} \) |
| 43 | \( 1 + 8.89T + 43T^{2} \) |
| 47 | \( 1 + 2.44T + 47T^{2} \) |
| 53 | \( 1 + 4.34T + 53T^{2} \) |
| 59 | \( 1 + 1.89T + 59T^{2} \) |
| 61 | \( 1 + 5.34T + 61T^{2} \) |
| 67 | \( 1 - 1.44T + 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 - 9.34T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−8.027086375797784856924669577004, −7.38669223646632197481899076236, −6.49866880258402391661377494326, −5.94248484526938516102918064744, −5.16117494249830695247356575562, −4.02835155466474128318589351458, −3.39225186097387603644406133265, −2.72600835508057881818660331510, −1.26222459211938058750437967144, 0,
1.26222459211938058750437967144, 2.72600835508057881818660331510, 3.39225186097387603644406133265, 4.02835155466474128318589351458, 5.16117494249830695247356575562, 5.94248484526938516102918064744, 6.49866880258402391661377494326, 7.38669223646632197481899076236, 8.027086375797784856924669577004
