| L(s) = 1 | − 7-s + 0.449·11-s − 3.89·13-s + 4.89·17-s − 5.89·19-s + 4.44·23-s + 0.449·29-s + 6·31-s − 0.101·37-s − 9.34·41-s + 6·43-s − 4.89·47-s − 6·49-s − 4.44·53-s − 4.89·59-s + 8.79·61-s + 14.7·67-s − 11.5·71-s + 3.89·73-s − 0.449·77-s − 3.89·79-s + 7.55·83-s − 12·89-s + 3.89·91-s − 15.8·97-s − 7.10·101-s − 10.7·103-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.135·11-s − 1.08·13-s + 1.18·17-s − 1.35·19-s + 0.927·23-s + 0.0834·29-s + 1.07·31-s − 0.0166·37-s − 1.45·41-s + 0.914·43-s − 0.714·47-s − 0.857·49-s − 0.611·53-s − 0.637·59-s + 1.12·61-s + 1.80·67-s − 1.37·71-s + 0.456·73-s − 0.0512·77-s − 0.438·79-s + 0.828·83-s − 1.27·89-s + 0.408·91-s − 1.61·97-s − 0.706·101-s − 1.06·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 0.449T + 11T^{2} \) |
| 13 | \( 1 + 3.89T + 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 + 5.89T + 19T^{2} \) |
| 23 | \( 1 - 4.44T + 23T^{2} \) |
| 29 | \( 1 - 0.449T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 0.101T + 37T^{2} \) |
| 41 | \( 1 + 9.34T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + 4.89T + 47T^{2} \) |
| 53 | \( 1 + 4.44T + 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 - 8.79T + 61T^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 - 3.89T + 73T^{2} \) |
| 79 | \( 1 + 3.89T + 79T^{2} \) |
| 83 | \( 1 - 7.55T + 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−7.963459057594579122403119533656, −6.90653048493958958531445178052, −6.60335726409188197834078815252, −5.58229840793489948713227592113, −4.92810387794930377824471525324, −4.13426275597960821572306435630, −3.18094941228393760994396172663, −2.48212152249939269578907577761, −1.32513231348801711207744572949, 0,
1.32513231348801711207744572949, 2.48212152249939269578907577761, 3.18094941228393760994396172663, 4.13426275597960821572306435630, 4.92810387794930377824471525324, 5.58229840793489948713227592113, 6.60335726409188197834078815252, 6.90653048493958958531445178052, 7.963459057594579122403119533656
