| L(s) = 1 | + 0.523·3-s − 5-s − 0.476·7-s − 2.72·9-s + 1.67·11-s + 2.67·13-s − 0.523·15-s + 1.67·17-s − 7.92·19-s − 0.249·21-s − 23-s + 25-s − 3·27-s − 2.20·29-s − 6.77·31-s + 0.878·33-s + 0.476·35-s + 4·37-s + 1.40·39-s + 5.97·41-s + 0.402·43-s + 2.72·45-s − 1.79·47-s − 6.77·49-s + 0.878·51-s − 10.8·53-s − 1.67·55-s + ⋯ |
| L(s) = 1 | + 0.302·3-s − 0.447·5-s − 0.179·7-s − 0.908·9-s + 0.505·11-s + 0.742·13-s − 0.135·15-s + 0.406·17-s − 1.81·19-s − 0.0544·21-s − 0.208·23-s + 0.200·25-s − 0.577·27-s − 0.408·29-s − 1.21·31-s + 0.153·33-s + 0.0804·35-s + 0.657·37-s + 0.224·39-s + 0.933·41-s + 0.0614·43-s + 0.406·45-s − 0.262·47-s − 0.967·49-s + 0.123·51-s − 1.48·53-s − 0.226·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 0.523T + 3T^{2} \) |
| 7 | \( 1 + 0.476T + 7T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 - 2.67T + 13T^{2} \) |
| 17 | \( 1 - 1.67T + 17T^{2} \) |
| 19 | \( 1 + 7.92T + 19T^{2} \) |
| 29 | \( 1 + 2.20T + 29T^{2} \) |
| 31 | \( 1 + 6.77T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 5.97T + 41T^{2} \) |
| 43 | \( 1 - 0.402T + 43T^{2} \) |
| 47 | \( 1 + 1.79T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 9.45T + 59T^{2} \) |
| 61 | \( 1 - 6.32T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 + 9.97T + 71T^{2} \) |
| 73 | \( 1 + 6.10T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4.49T + 83T^{2} \) |
| 89 | \( 1 + 3.59T + 89T^{2} \) |
| 97 | \( 1 + 6.08T + 97T^{2} \) |
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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
−8.831509683789441295958427503674, −8.169610470247004261151053069330, −7.41956921838051250455418890683, −6.29817554689136358218635620268, −5.87338081832233059754996266254, −4.57077346435066343333511912803, −3.76723187475248474017651227626, −2.93381683845095450530871213637, −1.70809905523468441561561219361, 0,
1.70809905523468441561561219361, 2.93381683845095450530871213637, 3.76723187475248474017651227626, 4.57077346435066343333511912803, 5.87338081832233059754996266254, 6.29817554689136358218635620268, 7.41956921838051250455418890683, 8.169610470247004261151053069330, 8.831509683789441295958427503674
