| L(s) = 1 | − 0.302·2-s + 3-s − 1.90·4-s + 3.34·5-s − 0.302·6-s + 0.428·7-s + 1.18·8-s + 9-s − 1.01·10-s + 4.01·11-s − 1.90·12-s + 5.31·13-s − 0.129·14-s + 3.34·15-s + 3.45·16-s + 1.05·17-s − 0.302·18-s − 3.12·19-s − 6.39·20-s + 0.428·21-s − 1.21·22-s + 23-s + 1.18·24-s + 6.22·25-s − 1.60·26-s + 27-s − 0.817·28-s + ⋯ |
| L(s) = 1 | − 0.214·2-s + 0.577·3-s − 0.954·4-s + 1.49·5-s − 0.123·6-s + 0.161·7-s + 0.418·8-s + 0.333·9-s − 0.320·10-s + 1.21·11-s − 0.550·12-s + 1.47·13-s − 0.0346·14-s + 0.864·15-s + 0.864·16-s + 0.254·17-s − 0.0713·18-s − 0.716·19-s − 1.42·20-s + 0.0934·21-s − 0.259·22-s + 0.208·23-s + 0.241·24-s + 1.24·25-s − 0.315·26-s + 0.192·27-s − 0.154·28-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
| \(L(1)\) |
\(\approx\) |
\(2.458690307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.458690307\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.302T + 2T^{2} \) |
| 5 | \( 1 - 3.34T + 5T^{2} \) |
| 7 | \( 1 - 0.428T + 7T^{2} \) |
| 11 | \( 1 - 4.01T + 11T^{2} \) |
| 13 | \( 1 - 5.31T + 13T^{2} \) |
| 17 | \( 1 - 1.05T + 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 - 7.14T + 41T^{2} \) |
| 43 | \( 1 - 3.22T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + 9.70T + 53T^{2} \) |
| 59 | \( 1 + 4.68T + 59T^{2} \) |
| 61 | \( 1 + 4.20T + 61T^{2} \) |
| 67 | \( 1 + 3.89T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 - 8.38T + 73T^{2} \) |
| 79 | \( 1 - 8.80T + 79T^{2} \) |
| 83 | \( 1 - 7.18T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 + 3.31T + 97T^{2} \) |
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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
−9.146589826705809583394245684743, −8.748262205294267577014415531609, −7.82433522310567660218820781017, −6.67852011342506284922386378673, −5.98490989669662509065309904493, −5.20923494724239873788094274213, −4.10159295216845121906982762897, −3.42914890560782025819935858006, −1.93100420637295074867136751316, −1.22279830034674351542939807868,
1.22279830034674351542939807868, 1.93100420637295074867136751316, 3.42914890560782025819935858006, 4.10159295216845121906982762897, 5.20923494724239873788094274213, 5.98490989669662509065309904493, 6.67852011342506284922386378673, 7.82433522310567660218820781017, 8.748262205294267577014415531609, 9.146589826705809583394245684743
