| L(s) = 1 | + 3-s + 5-s − 0.720·7-s + 9-s − 3.02·11-s − 2.07·13-s + 15-s − 1.71·17-s − 6.48·19-s − 0.720·21-s + 4.56·23-s + 25-s + 27-s + 1.21·29-s + 2.76·31-s − 3.02·33-s − 0.720·35-s + 2.56·37-s − 2.07·39-s − 3.40·41-s − 0.778·43-s + 45-s + 0.756·47-s − 6.48·49-s − 1.71·51-s − 10.9·53-s − 3.02·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.272·7-s + 0.333·9-s − 0.912·11-s − 0.576·13-s + 0.258·15-s − 0.415·17-s − 1.48·19-s − 0.157·21-s + 0.952·23-s + 0.200·25-s + 0.192·27-s + 0.225·29-s + 0.496·31-s − 0.526·33-s − 0.121·35-s + 0.421·37-s − 0.333·39-s − 0.532·41-s − 0.118·43-s + 0.149·45-s + 0.110·47-s − 0.925·49-s − 0.239·51-s − 1.51·53-s − 0.408·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| good | 7 | \( 1 + 0.720T + 7T^{2} \) |
| 11 | \( 1 + 3.02T + 11T^{2} \) |
| 13 | \( 1 + 2.07T + 13T^{2} \) |
| 17 | \( 1 + 1.71T + 17T^{2} \) |
| 19 | \( 1 + 6.48T + 19T^{2} \) |
| 23 | \( 1 - 4.56T + 23T^{2} \) |
| 29 | \( 1 - 1.21T + 29T^{2} \) |
| 31 | \( 1 - 2.76T + 31T^{2} \) |
| 37 | \( 1 - 2.56T + 37T^{2} \) |
| 41 | \( 1 + 3.40T + 41T^{2} \) |
| 43 | \( 1 + 0.778T + 43T^{2} \) |
| 47 | \( 1 - 0.756T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 - 0.542T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 7.98T + 73T^{2} \) |
| 79 | \( 1 - 1.13T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 + 5.03T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−8.154126585864317599946931217435, −7.39542827712045314163763227813, −6.62732343280857504249655453009, −5.96658824870015030293638655549, −4.89957434449436799083466907763, −4.41208848204929580978491828162, −3.14003103802886463187673986252, −2.58111642265775313428151711892, −1.63628969537079703947121134881, 0,
1.63628969537079703947121134881, 2.58111642265775313428151711892, 3.14003103802886463187673986252, 4.41208848204929580978491828162, 4.89957434449436799083466907763, 5.96658824870015030293638655549, 6.62732343280857504249655453009, 7.39542827712045314163763227813, 8.154126585864317599946931217435
