| L(s) = 1 | − 3-s + 1.53·5-s + 1.95·7-s + 9-s − 2.15·11-s − 0.0320·13-s − 1.53·15-s − 2.27·17-s − 5.52·19-s − 1.95·21-s + 6.90·23-s − 2.65·25-s − 27-s − 2.80·29-s + 1.86·31-s + 2.15·33-s + 2.99·35-s − 7.30·37-s + 0.0320·39-s − 11.8·41-s + 1.12·43-s + 1.53·45-s − 0.929·47-s − 3.18·49-s + 2.27·51-s − 11.1·53-s − 3.29·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.684·5-s + 0.738·7-s + 0.333·9-s − 0.648·11-s − 0.00888·13-s − 0.395·15-s − 0.552·17-s − 1.26·19-s − 0.426·21-s + 1.43·23-s − 0.531·25-s − 0.192·27-s − 0.520·29-s + 0.334·31-s + 0.374·33-s + 0.505·35-s − 1.20·37-s + 0.00513·39-s − 1.84·41-s + 0.171·43-s + 0.228·45-s − 0.135·47-s − 0.454·49-s + 0.319·51-s − 1.52·53-s − 0.444·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 1.53T + 5T^{2} \) |
| 7 | \( 1 - 1.95T + 7T^{2} \) |
| 11 | \( 1 + 2.15T + 11T^{2} \) |
| 13 | \( 1 + 0.0320T + 13T^{2} \) |
| 17 | \( 1 + 2.27T + 17T^{2} \) |
| 19 | \( 1 + 5.52T + 19T^{2} \) |
| 23 | \( 1 - 6.90T + 23T^{2} \) |
| 29 | \( 1 + 2.80T + 29T^{2} \) |
| 31 | \( 1 - 1.86T + 31T^{2} \) |
| 37 | \( 1 + 7.30T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 1.12T + 43T^{2} \) |
| 47 | \( 1 + 0.929T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 - 8.11T + 59T^{2} \) |
| 61 | \( 1 + 7.44T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 7.98T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 5.86T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 - 7.24T + 89T^{2} \) |
| 97 | \( 1 - 2.96T + 97T^{2} \) |
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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
−8.207200267676954168212287875459, −7.19383502582399522574686734349, −6.60147336185156574765641162103, −5.79955433681207283752074575756, −5.05038908461551548065041314041, −4.59002259352377360914231498640, −3.41000186456630771158030012682, −2.24210525531034922227719843401, −1.53344292590671643553213393521, 0,
1.53344292590671643553213393521, 2.24210525531034922227719843401, 3.41000186456630771158030012682, 4.59002259352377360914231498640, 5.05038908461551548065041314041, 5.79955433681207283752074575756, 6.60147336185156574765641162103, 7.19383502582399522574686734349, 8.207200267676954168212287875459
