| L(s) = 1 | − 2.68·2-s − 3-s + 5.20·4-s + 1.06·5-s + 2.68·6-s + 1.62·7-s − 8.60·8-s + 9-s − 2.84·10-s − 4.08·11-s − 5.20·12-s + 2.36·13-s − 4.35·14-s − 1.06·15-s + 12.6·16-s + 1.98·17-s − 2.68·18-s + 1.29·19-s + 5.52·20-s − 1.62·21-s + 10.9·22-s − 23-s + 8.60·24-s − 3.87·25-s − 6.33·26-s − 27-s + 8.44·28-s + ⋯ |
| L(s) = 1 | − 1.89·2-s − 0.577·3-s + 2.60·4-s + 0.474·5-s + 1.09·6-s + 0.612·7-s − 3.04·8-s + 0.333·9-s − 0.901·10-s − 1.23·11-s − 1.50·12-s + 0.654·13-s − 1.16·14-s − 0.274·15-s + 3.17·16-s + 0.480·17-s − 0.632·18-s + 0.297·19-s + 1.23·20-s − 0.353·21-s + 2.33·22-s − 0.208·23-s + 1.75·24-s − 0.774·25-s − 1.24·26-s − 0.192·27-s + 1.59·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.68T + 2T^{2} \) |
| 5 | \( 1 - 1.06T + 5T^{2} \) |
| 7 | \( 1 - 1.62T + 7T^{2} \) |
| 11 | \( 1 + 4.08T + 11T^{2} \) |
| 13 | \( 1 - 2.36T + 13T^{2} \) |
| 17 | \( 1 - 1.98T + 17T^{2} \) |
| 19 | \( 1 - 1.29T + 19T^{2} \) |
| 31 | \( 1 + 0.969T + 31T^{2} \) |
| 37 | \( 1 + 4.37T + 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 + 4.54T + 43T^{2} \) |
| 47 | \( 1 - 0.146T + 47T^{2} \) |
| 53 | \( 1 - 3.30T + 53T^{2} \) |
| 59 | \( 1 + 0.410T + 59T^{2} \) |
| 61 | \( 1 - 6.07T + 61T^{2} \) |
| 67 | \( 1 + 1.92T + 67T^{2} \) |
| 71 | \( 1 + 0.306T + 71T^{2} \) |
| 73 | \( 1 - 1.40T + 73T^{2} \) |
| 79 | \( 1 + 7.54T + 79T^{2} \) |
| 83 | \( 1 - 4.19T + 83T^{2} \) |
| 89 | \( 1 + 4.44T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−8.681847717193184034734223036207, −8.118021104396554290007892417634, −7.46971057352090243272327297061, −6.65235821031313702847956860716, −5.78748700238732083713404178962, −5.11608689701584979752668585002, −3.38996442296505535305098819871, −2.15864619216103618272038038108, −1.37271025165194484008859244780, 0,
1.37271025165194484008859244780, 2.15864619216103618272038038108, 3.38996442296505535305098819871, 5.11608689701584979752668585002, 5.78748700238732083713404178962, 6.65235821031313702847956860716, 7.46971057352090243272327297061, 8.118021104396554290007892417634, 8.681847717193184034734223036207
