| L(s) = 1 | + 1.77·2-s − 1.73·3-s + 1.15·4-s + 0.320·5-s − 3.07·6-s − 4.28·7-s − 1.50·8-s + 0.00809·9-s + 0.569·10-s − 4.70·11-s − 1.99·12-s − 6.93·13-s − 7.60·14-s − 0.556·15-s − 4.97·16-s − 1.58·17-s + 0.0143·18-s + 19-s + 0.369·20-s + 7.42·21-s − 8.34·22-s + 6.09·23-s + 2.61·24-s − 4.89·25-s − 12.3·26-s + 5.18·27-s − 4.93·28-s + ⋯ |
| L(s) = 1 | + 1.25·2-s − 1.00·3-s + 0.576·4-s + 0.143·5-s − 1.25·6-s − 1.61·7-s − 0.532·8-s + 0.00269·9-s + 0.180·10-s − 1.41·11-s − 0.576·12-s − 1.92·13-s − 2.03·14-s − 0.143·15-s − 1.24·16-s − 0.383·17-s + 0.00338·18-s + 0.229·19-s + 0.0826·20-s + 1.62·21-s − 1.78·22-s + 1.27·23-s + 0.532·24-s − 0.979·25-s − 2.41·26-s + 0.998·27-s − 0.932·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.002219444669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.002219444669\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 - T \) |
| 317 | \( 1 + T \) |
| good | 2 | \( 1 - 1.77T + 2T^{2} \) |
| 3 | \( 1 + 1.73T + 3T^{2} \) |
| 5 | \( 1 - 0.320T + 5T^{2} \) |
| 7 | \( 1 + 4.28T + 7T^{2} \) |
| 11 | \( 1 + 4.70T + 11T^{2} \) |
| 13 | \( 1 + 6.93T + 13T^{2} \) |
| 17 | \( 1 + 1.58T + 17T^{2} \) |
| 23 | \( 1 - 6.09T + 23T^{2} \) |
| 29 | \( 1 + 7.91T + 29T^{2} \) |
| 31 | \( 1 + 1.68T + 31T^{2} \) |
| 37 | \( 1 + 4.36T + 37T^{2} \) |
| 41 | \( 1 - 5.75T + 41T^{2} \) |
| 43 | \( 1 + 2.46T + 43T^{2} \) |
| 47 | \( 1 + 4.33T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 0.199T + 59T^{2} \) |
| 61 | \( 1 - 0.396T + 61T^{2} \) |
| 67 | \( 1 - 6.23T + 67T^{2} \) |
| 71 | \( 1 + 0.427T + 71T^{2} \) |
| 73 | \( 1 - 7.10T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 + 8.44T + 83T^{2} \) |
| 89 | \( 1 - 17.1T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−7.69716435704862886792669351685, −7.05408726797976591582356864402, −6.41306089474360625088458225826, −5.71807769830969591200621248554, −5.20384720929926819299839585797, −4.79032052882085553256136482759, −3.66543060785135720141528791019, −2.90521327315410971433791640255, −2.37164582385666767394506934076, −0.01927947957304950154390232833,
0.01927947957304950154390232833, 2.37164582385666767394506934076, 2.90521327315410971433791640255, 3.66543060785135720141528791019, 4.79032052882085553256136482759, 5.20384720929926819299839585797, 5.71807769830969591200621248554, 6.41306089474360625088458225826, 7.05408726797976591582356864402, 7.69716435704862886792669351685
