| L(s) = 1 | − 2·3-s + 5-s − 7-s + 9-s − 11-s + 13-s − 2·15-s − 2·17-s − 5·19-s + 2·21-s + 9·23-s − 4·25-s + 4·27-s + 7·29-s − 31-s + 2·33-s − 35-s − 6·37-s − 2·39-s + 10·41-s − 11·43-s + 45-s − 7·47-s + 49-s + 4·51-s + 3·53-s − 55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s − 0.516·15-s − 0.485·17-s − 1.14·19-s + 0.436·21-s + 1.87·23-s − 4/5·25-s + 0.769·27-s + 1.29·29-s − 0.179·31-s + 0.348·33-s − 0.169·35-s − 0.986·37-s − 0.320·39-s + 1.56·41-s − 1.67·43-s + 0.149·45-s − 1.02·47-s + 1/7·49-s + 0.560·51-s + 0.412·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−7.20128144818778769931133556615, −6.53795516228731029091084372163, −6.26907950901898273700166858662, −5.31535279339691544760991147641, −4.96641763118445740319855030469, −4.04530329623052968565212186337, −3.04958451374052878960738897544, −2.19867671604621791423116385588, −1.04408567521552724379090988190, 0,
1.04408567521552724379090988190, 2.19867671604621791423116385588, 3.04958451374052878960738897544, 4.04530329623052968565212186337, 4.96641763118445740319855030469, 5.31535279339691544760991147641, 6.26907950901898273700166858662, 6.53795516228731029091084372163, 7.20128144818778769931133556615
