| L(s) = 1 | + 1.15·2-s − 3-s − 0.669·4-s − 1.15·6-s − 7-s − 3.07·8-s + 9-s − 3.62·11-s + 0.669·12-s − 1.69·13-s − 1.15·14-s − 2.21·16-s + 17-s + 1.15·18-s + 4.68·19-s + 21-s − 4.18·22-s − 6.53·23-s + 3.07·24-s − 1.95·26-s − 27-s + 0.669·28-s + 7.19·29-s + 8.65·31-s + 3.60·32-s + 3.62·33-s + 1.15·34-s + ⋯ |
| L(s) = 1 | + 0.815·2-s − 0.577·3-s − 0.334·4-s − 0.470·6-s − 0.377·7-s − 1.08·8-s + 0.333·9-s − 1.09·11-s + 0.193·12-s − 0.469·13-s − 0.308·14-s − 0.553·16-s + 0.242·17-s + 0.271·18-s + 1.07·19-s + 0.218·21-s − 0.892·22-s − 1.36·23-s + 0.628·24-s − 0.382·26-s − 0.192·27-s + 0.126·28-s + 1.33·29-s + 1.55·31-s + 0.637·32-s + 0.631·33-s + 0.197·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - 1.15T + 2T^{2} \) |
| 11 | \( 1 + 3.62T + 11T^{2} \) |
| 13 | \( 1 + 1.69T + 13T^{2} \) |
| 19 | \( 1 - 4.68T + 19T^{2} \) |
| 23 | \( 1 + 6.53T + 23T^{2} \) |
| 29 | \( 1 - 7.19T + 29T^{2} \) |
| 31 | \( 1 - 8.65T + 31T^{2} \) |
| 37 | \( 1 - 7.64T + 37T^{2} \) |
| 41 | \( 1 - 7.06T + 41T^{2} \) |
| 43 | \( 1 - 6.12T + 43T^{2} \) |
| 47 | \( 1 - 8.93T + 47T^{2} \) |
| 53 | \( 1 + 9.22T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 + 7.44T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + 4.52T + 83T^{2} \) |
| 89 | \( 1 + 8.78T + 89T^{2} \) |
| 97 | \( 1 + 0.773T + 97T^{2} \) |
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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
−7.43591763195621513007608749944, −6.38216637881303315925887004449, −5.85235365640584792872874946191, −5.41190991309337426700734117150, −4.41530909284753393729636229229, −4.30078053853130658607819987277, −2.90639307342141066115697905684, −2.70857677231952578070789131576, −1.04092561250382355655400728296, 0,
1.04092561250382355655400728296, 2.70857677231952578070789131576, 2.90639307342141066115697905684, 4.30078053853130658607819987277, 4.41530909284753393729636229229, 5.41190991309337426700734117150, 5.85235365640584792872874946191, 6.38216637881303315925887004449, 7.43591763195621513007608749944
