| L(s) = 1 | − 2.00·2-s + 3-s + 2.01·4-s + 2.86·5-s − 2.00·6-s − 1.37·7-s − 0.0389·8-s + 9-s − 5.74·10-s + 4.45·11-s + 2.01·12-s − 0.546·13-s + 2.76·14-s + 2.86·15-s − 3.96·16-s − 17-s − 2.00·18-s − 5.48·19-s + 5.78·20-s − 1.37·21-s − 8.93·22-s − 5.85·23-s − 0.0389·24-s + 3.21·25-s + 1.09·26-s + 27-s − 2.78·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 1.00·4-s + 1.28·5-s − 0.818·6-s − 0.520·7-s − 0.0137·8-s + 0.333·9-s − 1.81·10-s + 1.34·11-s + 0.582·12-s − 0.151·13-s + 0.737·14-s + 0.740·15-s − 0.990·16-s − 0.242·17-s − 0.472·18-s − 1.25·19-s + 1.29·20-s − 0.300·21-s − 1.90·22-s − 1.22·23-s − 0.00795·24-s + 0.643·25-s + 0.214·26-s + 0.192·27-s − 0.525·28-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 | \( 1 + 2.00T + 2T^{2} \) |
| 5 | \( 1 - 2.86T + 5T^{2} \) |
| 7 | \( 1 + 1.37T + 7T^{2} \) |
| 11 | \( 1 - 4.45T + 11T^{2} \) |
| 13 | \( 1 + 0.546T + 13T^{2} \) |
| 19 | \( 1 + 5.48T + 19T^{2} \) |
| 23 | \( 1 + 5.85T + 23T^{2} \) |
| 29 | \( 1 - 1.43T + 29T^{2} \) |
| 31 | \( 1 + 6.52T + 31T^{2} \) |
| 37 | \( 1 - 2.12T + 37T^{2} \) |
| 41 | \( 1 - 4.65T + 41T^{2} \) |
| 43 | \( 1 + 3.60T + 43T^{2} \) |
| 47 | \( 1 + 3.40T + 47T^{2} \) |
| 53 | \( 1 + 3.06T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 + 4.18T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 1.13T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 4.80T + 79T^{2} \) |
| 83 | \( 1 - 2.37T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−7.63852040680868516918774480569, −6.83795356367686902995138959499, −6.39398530972969482002793546909, −5.75138129119930579241219211172, −4.50895032482123088147454225295, −3.85470456381661045498580560610, −2.67708065840840844999346487314, −1.90396710847812961701037034559, −1.42307826119273183385881922342, 0,
1.42307826119273183385881922342, 1.90396710847812961701037034559, 2.67708065840840844999346487314, 3.85470456381661045498580560610, 4.50895032482123088147454225295, 5.75138129119930579241219211172, 6.39398530972969482002793546909, 6.83795356367686902995138959499, 7.63852040680868516918774480569
