| L(s) = 1 | − 1.97·2-s + 3-s + 1.90·4-s − 3.88·5-s − 1.97·6-s + 2.64·7-s + 0.195·8-s + 9-s + 7.67·10-s + 4.76·11-s + 1.90·12-s + 3.33·13-s − 5.22·14-s − 3.88·15-s − 4.18·16-s − 7.28·17-s − 1.97·18-s + 5.43·19-s − 7.38·20-s + 2.64·21-s − 9.41·22-s + 23-s + 0.195·24-s + 10.0·25-s − 6.57·26-s + 27-s + 5.02·28-s + ⋯ |
| L(s) = 1 | − 1.39·2-s + 0.577·3-s + 0.950·4-s − 1.73·5-s − 0.806·6-s + 0.999·7-s + 0.0692·8-s + 0.333·9-s + 2.42·10-s + 1.43·11-s + 0.548·12-s + 0.923·13-s − 1.39·14-s − 1.00·15-s − 1.04·16-s − 1.76·17-s − 0.465·18-s + 1.24·19-s − 1.65·20-s + 0.576·21-s − 2.00·22-s + 0.208·23-s + 0.0399·24-s + 2.01·25-s − 1.29·26-s + 0.192·27-s + 0.949·28-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
| \(L(1)\) |
\(\approx\) |
\(0.9304866719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9304866719\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.97T + 2T^{2} \) |
| 5 | \( 1 + 3.88T + 5T^{2} \) |
| 7 | \( 1 - 2.64T + 7T^{2} \) |
| 11 | \( 1 - 4.76T + 11T^{2} \) |
| 13 | \( 1 - 3.33T + 13T^{2} \) |
| 17 | \( 1 + 7.28T + 17T^{2} \) |
| 19 | \( 1 - 5.43T + 19T^{2} \) |
| 31 | \( 1 - 6.87T + 31T^{2} \) |
| 37 | \( 1 + 9.05T + 37T^{2} \) |
| 41 | \( 1 - 5.39T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 8.49T + 47T^{2} \) |
| 53 | \( 1 + 7.62T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 0.452T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 - 6.12T + 73T^{2} \) |
| 79 | \( 1 + 9.83T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 + 0.490T + 89T^{2} \) |
| 97 | \( 1 + 0.789T + 97T^{2} \) |
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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
−8.824185141263601307409211951148, −8.481250081584119843059767311955, −7.942266745371610503937428895550, −7.11442719670836749291728286854, −6.61303002560571956975047548899, −4.74505845240414890333225566204, −4.19383684933417656190266914534, −3.28863004058053548183523864267, −1.75720142666994672631958217711, −0.826454854217917746203835466043,
0.826454854217917746203835466043, 1.75720142666994672631958217711, 3.28863004058053548183523864267, 4.19383684933417656190266914534, 4.74505845240414890333225566204, 6.61303002560571956975047548899, 7.11442719670836749291728286854, 7.942266745371610503937428895550, 8.481250081584119843059767311955, 8.824185141263601307409211951148
