| L(s) = 1 | + 2.05·2-s + 3-s + 2.21·4-s + 2.38·5-s + 2.05·6-s − 5.11·7-s + 0.447·8-s + 9-s + 4.90·10-s + 1.96·11-s + 2.21·12-s − 2.07·13-s − 10.4·14-s + 2.38·15-s − 3.51·16-s − 17-s + 2.05·18-s − 2.46·19-s + 5.29·20-s − 5.11·21-s + 4.03·22-s − 8.81·23-s + 0.447·24-s + 0.695·25-s − 4.26·26-s + 27-s − 11.3·28-s + ⋯ |
| L(s) = 1 | + 1.45·2-s + 0.577·3-s + 1.10·4-s + 1.06·5-s + 0.838·6-s − 1.93·7-s + 0.158·8-s + 0.333·9-s + 1.54·10-s + 0.591·11-s + 0.640·12-s − 0.576·13-s − 2.80·14-s + 0.616·15-s − 0.879·16-s − 0.242·17-s + 0.484·18-s − 0.565·19-s + 1.18·20-s − 1.11·21-s + 0.859·22-s − 1.83·23-s + 0.0912·24-s + 0.139·25-s − 0.837·26-s + 0.192·27-s − 2.14·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 - 2.05T + 2T^{2} \) |
| 5 | \( 1 - 2.38T + 5T^{2} \) |
| 7 | \( 1 + 5.11T + 7T^{2} \) |
| 11 | \( 1 - 1.96T + 11T^{2} \) |
| 13 | \( 1 + 2.07T + 13T^{2} \) |
| 19 | \( 1 + 2.46T + 19T^{2} \) |
| 23 | \( 1 + 8.81T + 23T^{2} \) |
| 29 | \( 1 - 5.45T + 29T^{2} \) |
| 31 | \( 1 - 5.62T + 31T^{2} \) |
| 37 | \( 1 + 8.11T + 37T^{2} \) |
| 41 | \( 1 + 0.208T + 41T^{2} \) |
| 43 | \( 1 + 5.51T + 43T^{2} \) |
| 47 | \( 1 - 7.50T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 5.74T + 59T^{2} \) |
| 61 | \( 1 + 0.000881T + 61T^{2} \) |
| 67 | \( 1 + 5.07T + 67T^{2} \) |
| 71 | \( 1 - 7.80T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 - 2.62T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 3.16T + 97T^{2} \) |
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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
−7.01285387119677244394454641486, −6.44866703044705940439447064443, −6.19952539477518182347000806223, −5.48113604129363828722286489516, −4.48954997527002069608665006755, −3.90506798942182058137160987955, −3.13999514869538994849199755068, −2.59268146538593279754196843955, −1.82211216960878127383327064056, 0,
1.82211216960878127383327064056, 2.59268146538593279754196843955, 3.13999514869538994849199755068, 3.90506798942182058137160987955, 4.48954997527002069608665006755, 5.48113604129363828722286489516, 6.19952539477518182347000806223, 6.44866703044705940439447064443, 7.01285387119677244394454641486
