| L(s) = 1 | − 2.30·3-s + 5-s − 3.30·7-s + 2.30·9-s + 2.60·11-s − 2.30·13-s − 2.30·15-s + 1.30·17-s + 0.605·19-s + 7.60·21-s + 6.90·23-s + 25-s + 1.60·27-s − 29-s − 6.69·31-s − 6·33-s − 3.30·35-s − 0.605·37-s + 5.30·39-s + 8.60·41-s − 3.30·43-s + 2.30·45-s − 5.21·47-s + 3.90·49-s − 3·51-s − 13.3·53-s + 2.60·55-s + ⋯ |
| L(s) = 1 | − 1.32·3-s + 0.447·5-s − 1.24·7-s + 0.767·9-s + 0.785·11-s − 0.638·13-s − 0.594·15-s + 0.315·17-s + 0.138·19-s + 1.65·21-s + 1.44·23-s + 0.200·25-s + 0.308·27-s − 0.185·29-s − 1.20·31-s − 1.04·33-s − 0.558·35-s − 0.0995·37-s + 0.849·39-s + 1.34·41-s − 0.503·43-s + 0.343·45-s − 0.760·47-s + 0.558·49-s − 0.420·51-s − 1.82·53-s + 0.351·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 2.30T + 3T^{2} \) |
| 7 | \( 1 + 3.30T + 7T^{2} \) |
| 11 | \( 1 - 2.60T + 11T^{2} \) |
| 13 | \( 1 + 2.30T + 13T^{2} \) |
| 17 | \( 1 - 1.30T + 17T^{2} \) |
| 19 | \( 1 - 0.605T + 19T^{2} \) |
| 23 | \( 1 - 6.90T + 23T^{2} \) |
| 31 | \( 1 + 6.69T + 31T^{2} \) |
| 37 | \( 1 + 0.605T + 37T^{2} \) |
| 41 | \( 1 - 8.60T + 41T^{2} \) |
| 43 | \( 1 + 3.30T + 43T^{2} \) |
| 47 | \( 1 + 5.21T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 - 7.69T + 59T^{2} \) |
| 61 | \( 1 - 0.302T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 5.21T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 8.90T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 - 7.81T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−8.923791757366169321337633429384, −7.55326736257573072920879806230, −6.73640664366581527406235082070, −6.34871134837161661656754453653, −5.50912553622766808907864299605, −4.89026245062592187852389308560, −3.72224555384496533398398996051, −2.76666931204636442334554366348, −1.26261605274011305808623050539, 0,
1.26261605274011305808623050539, 2.76666931204636442334554366348, 3.72224555384496533398398996051, 4.89026245062592187852389308560, 5.50912553622766808907864299605, 6.34871134837161661656754453653, 6.73640664366581527406235082070, 7.55326736257573072920879806230, 8.923791757366169321337633429384
