| L(s) = 1 | − 1.99·3-s + 2.83·5-s + 3.85·7-s + 0.991·9-s − 6.23·13-s − 5.65·15-s − 4.97·17-s + 19-s − 7.69·21-s − 6.56·23-s + 3.01·25-s + 4.01·27-s + 3.54·29-s + 6.81·31-s + 10.8·35-s − 5.04·37-s + 12.4·39-s + 7.13·41-s + 11.4·43-s + 2.80·45-s − 9.88·47-s + 7.82·49-s + 9.94·51-s + 3.38·53-s − 1.99·57-s − 6.79·59-s − 13.7·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.26·5-s + 1.45·7-s + 0.330·9-s − 1.72·13-s − 1.46·15-s − 1.20·17-s + 0.229·19-s − 1.67·21-s − 1.36·23-s + 0.602·25-s + 0.772·27-s + 0.657·29-s + 1.22·31-s + 1.84·35-s − 0.830·37-s + 1.99·39-s + 1.11·41-s + 1.75·43-s + 0.418·45-s − 1.44·47-s + 1.11·49-s + 1.39·51-s + 0.464·53-s − 0.264·57-s − 0.884·59-s − 1.75·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.99T + 3T^{2} \) |
| 5 | \( 1 - 2.83T + 5T^{2} \) |
| 7 | \( 1 - 3.85T + 7T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 + 4.97T + 17T^{2} \) |
| 23 | \( 1 + 6.56T + 23T^{2} \) |
| 29 | \( 1 - 3.54T + 29T^{2} \) |
| 31 | \( 1 - 6.81T + 31T^{2} \) |
| 37 | \( 1 + 5.04T + 37T^{2} \) |
| 41 | \( 1 - 7.13T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 9.88T + 47T^{2} \) |
| 53 | \( 1 - 3.38T + 53T^{2} \) |
| 59 | \( 1 + 6.79T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 - 9.47T + 67T^{2} \) |
| 71 | \( 1 + 8.56T + 71T^{2} \) |
| 73 | \( 1 - 2.78T + 73T^{2} \) |
| 79 | \( 1 - 0.163T + 79T^{2} \) |
| 83 | \( 1 + 7.82T + 83T^{2} \) |
| 89 | \( 1 + 3.81T + 89T^{2} \) |
| 97 | \( 1 - 9.00T + 97T^{2} \) |
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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
−7.34735468373568826991414199552, −6.40500059740371313349983288649, −6.04027099033104632230853450986, −5.24794578363781510636215723363, −4.78873534197250652218163403656, −4.33494454958136065139257132173, −2.63334097959303876725016321932, −2.15385637570297665444390376466, −1.26236629297757188849412776875, 0,
1.26236629297757188849412776875, 2.15385637570297665444390376466, 2.63334097959303876725016321932, 4.33494454958136065139257132173, 4.78873534197250652218163403656, 5.24794578363781510636215723363, 6.04027099033104632230853450986, 6.40500059740371313349983288649, 7.34735468373568826991414199552
