| L(s) = 1 | − 1.81·3-s − 5-s + 4.91·7-s + 0.289·9-s − 0.578·11-s − 6.39·13-s + 1.81·15-s − 0.710·17-s − 19-s − 8.91·21-s + 2.71·23-s + 25-s + 4.91·27-s + 6.54·29-s − 1.42·31-s + 1.04·33-s − 4.91·35-s − 9.10·37-s + 11.5·39-s − 11.0·41-s − 5.83·43-s − 0.289·45-s + 1.15·47-s + 17.1·49-s + 1.28·51-s + 13.2·53-s + 0.578·55-s + ⋯ |
| L(s) = 1 | − 1.04·3-s − 0.447·5-s + 1.85·7-s + 0.0963·9-s − 0.174·11-s − 1.77·13-s + 0.468·15-s − 0.172·17-s − 0.229·19-s − 1.94·21-s + 0.565·23-s + 0.200·25-s + 0.946·27-s + 1.21·29-s − 0.255·31-s + 0.182·33-s − 0.831·35-s − 1.49·37-s + 1.85·39-s − 1.72·41-s − 0.889·43-s − 0.0431·45-s + 0.168·47-s + 2.45·49-s + 0.180·51-s + 1.82·53-s + 0.0779·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.81T + 3T^{2} \) |
| 7 | \( 1 - 4.91T + 7T^{2} \) |
| 11 | \( 1 + 0.578T + 11T^{2} \) |
| 13 | \( 1 + 6.39T + 13T^{2} \) |
| 17 | \( 1 + 0.710T + 17T^{2} \) |
| 23 | \( 1 - 2.71T + 23T^{2} \) |
| 29 | \( 1 - 6.54T + 29T^{2} \) |
| 31 | \( 1 + 1.42T + 31T^{2} \) |
| 37 | \( 1 + 9.10T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 + 5.83T + 43T^{2} \) |
| 47 | \( 1 - 1.15T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 9.04T + 61T^{2} \) |
| 67 | \( 1 + 2.97T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 9.38T + 73T^{2} \) |
| 79 | \( 1 + 4.37T + 79T^{2} \) |
| 83 | \( 1 - 0.372T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 - 3.94T + 97T^{2} \) |
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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
−8.860802365675483176211066258883, −8.254614497709732878824615339164, −7.38394875407072967812688464763, −6.76096379848190048134819747909, −5.41099517224066101967811834690, −4.99732562136610174913424466559, −4.40055200700155236134090076653, −2.78532423606824347924427876538, −1.54513954007592035166355729424, 0,
1.54513954007592035166355729424, 2.78532423606824347924427876538, 4.40055200700155236134090076653, 4.99732562136610174913424466559, 5.41099517224066101967811834690, 6.76096379848190048134819747909, 7.38394875407072967812688464763, 8.254614497709732878824615339164, 8.860802365675483176211066258883
