| L(s) = 1 | + 1.86·3-s − 1.86·5-s + 0.860·7-s + 0.462·9-s + 11-s − 3.78·13-s − 3.46·15-s − 3.72·17-s + 19-s + 1.60·21-s + 0.323·23-s − 1.53·25-s − 4.72·27-s + 8.18·29-s + 4.10·31-s + 1.86·33-s − 1.60·35-s + 2.60·37-s − 7.04·39-s − 10.4·41-s − 5.38·43-s − 0.860·45-s − 0.796·47-s − 6.25·49-s − 6.92·51-s − 2.92·53-s − 1.86·55-s + ⋯ |
| L(s) = 1 | + 1.07·3-s − 0.832·5-s + 0.325·7-s + 0.154·9-s + 0.301·11-s − 1.05·13-s − 0.894·15-s − 0.902·17-s + 0.229·19-s + 0.349·21-s + 0.0674·23-s − 0.307·25-s − 0.908·27-s + 1.51·29-s + 0.738·31-s + 0.323·33-s − 0.270·35-s + 0.427·37-s − 1.12·39-s − 1.62·41-s − 0.821·43-s − 0.128·45-s − 0.116·47-s − 0.894·49-s − 0.969·51-s − 0.401·53-s − 0.250·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.86T + 3T^{2} \) |
| 5 | \( 1 + 1.86T + 5T^{2} \) |
| 7 | \( 1 - 0.860T + 7T^{2} \) |
| 13 | \( 1 + 3.78T + 13T^{2} \) |
| 17 | \( 1 + 3.72T + 17T^{2} \) |
| 23 | \( 1 - 0.323T + 23T^{2} \) |
| 29 | \( 1 - 8.18T + 29T^{2} \) |
| 31 | \( 1 - 4.10T + 31T^{2} \) |
| 37 | \( 1 - 2.60T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 5.38T + 43T^{2} \) |
| 47 | \( 1 + 0.796T + 47T^{2} \) |
| 53 | \( 1 + 2.92T + 53T^{2} \) |
| 59 | \( 1 + 0.751T + 59T^{2} \) |
| 61 | \( 1 + 6.64T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 + 4.13T + 71T^{2} \) |
| 73 | \( 1 - 5.72T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 16.8T + 83T^{2} \) |
| 89 | \( 1 + 0.730T + 89T^{2} \) |
| 97 | \( 1 + 0.751T + 97T^{2} \) |
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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
−8.375928908877585005461785454983, −7.64471258429700450551257144468, −7.01025063476052120678470233586, −6.12067336988317441885015201964, −4.84473344504763201830955369903, −4.39107471273844852053366895686, −3.33300631734988120194837460608, −2.72046684190849323777623572499, −1.67739542852901017710678863099, 0,
1.67739542852901017710678863099, 2.72046684190849323777623572499, 3.33300631734988120194837460608, 4.39107471273844852053366895686, 4.84473344504763201830955369903, 6.12067336988317441885015201964, 7.01025063476052120678470233586, 7.64471258429700450551257144468, 8.375928908877585005461785454983
