| L(s) = 1 | − 0.904·2-s − 0.678·3-s − 1.18·4-s − 3.00·5-s + 0.613·6-s + 4.38·7-s + 2.87·8-s − 2.53·9-s + 2.72·10-s + 2.03·11-s + 0.801·12-s − 3.96·14-s + 2.04·15-s − 0.239·16-s + 4.92·17-s + 2.29·18-s + 6.06·19-s + 3.55·20-s − 2.97·21-s − 1.84·22-s + 1.39·23-s − 1.95·24-s + 4.04·25-s + 3.75·27-s − 5.17·28-s − 0.144·29-s − 1.84·30-s + ⋯ |
| L(s) = 1 | − 0.639·2-s − 0.391·3-s − 0.590·4-s − 1.34·5-s + 0.250·6-s + 1.65·7-s + 1.01·8-s − 0.846·9-s + 0.860·10-s + 0.614·11-s + 0.231·12-s − 1.05·14-s + 0.527·15-s − 0.0599·16-s + 1.19·17-s + 0.541·18-s + 1.39·19-s + 0.794·20-s − 0.648·21-s − 0.393·22-s + 0.290·23-s − 0.398·24-s + 0.809·25-s + 0.723·27-s − 0.978·28-s − 0.0268·29-s − 0.337·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.014236577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.014236577\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 0.904T + 2T^{2} \) |
| 3 | \( 1 + 0.678T + 3T^{2} \) |
| 5 | \( 1 + 3.00T + 5T^{2} \) |
| 7 | \( 1 - 4.38T + 7T^{2} \) |
| 11 | \( 1 - 2.03T + 11T^{2} \) |
| 17 | \( 1 - 4.92T + 17T^{2} \) |
| 19 | \( 1 - 6.06T + 19T^{2} \) |
| 23 | \( 1 - 1.39T + 23T^{2} \) |
| 29 | \( 1 + 0.144T + 29T^{2} \) |
| 37 | \( 1 - 7.22T + 37T^{2} \) |
| 41 | \( 1 - 5.56T + 41T^{2} \) |
| 43 | \( 1 - 0.0413T + 43T^{2} \) |
| 47 | \( 1 - 3.57T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 2.24T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 7.00T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 2.79T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 + 6.93T + 83T^{2} \) |
| 89 | \( 1 + 7.30T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−8.209034575294236576743569711450, −7.59266903409905017677814243495, −7.31254354714352159490525351738, −5.81895932974642745092948894548, −5.24524987584584944345363396454, −4.49642241218631773066467397250, −3.92490385093004978012665483566, −2.91155550056603999459537028966, −1.33981980322304942215890130454, −0.72537856436314177333018772430,
0.72537856436314177333018772430, 1.33981980322304942215890130454, 2.91155550056603999459537028966, 3.92490385093004978012665483566, 4.49642241218631773066467397250, 5.24524987584584944345363396454, 5.81895932974642745092948894548, 7.31254354714352159490525351738, 7.59266903409905017677814243495, 8.209034575294236576743569711450
