L(s) = 1 | − 2.56·2-s − 2.35·3-s + 4.57·4-s + 1.72·5-s + 6.03·6-s − 3.13·7-s − 6.61·8-s + 2.54·9-s − 4.43·10-s − 4.23·11-s − 10.7·12-s + 2.84·13-s + 8.03·14-s − 4.07·15-s + 7.80·16-s − 1.50·17-s − 6.51·18-s − 1.85·19-s + 7.91·20-s + 7.37·21-s + 10.8·22-s − 7.40·23-s + 15.5·24-s − 2.00·25-s − 7.30·26-s + 1.08·27-s − 14.3·28-s + ⋯ |
L(s) = 1 | − 1.81·2-s − 1.35·3-s + 2.28·4-s + 0.773·5-s + 2.46·6-s − 1.18·7-s − 2.33·8-s + 0.846·9-s − 1.40·10-s − 1.27·11-s − 3.11·12-s + 0.790·13-s + 2.14·14-s − 1.05·15-s + 1.95·16-s − 0.364·17-s − 1.53·18-s − 0.424·19-s + 1.77·20-s + 1.60·21-s + 2.31·22-s − 1.54·23-s + 3.17·24-s − 0.401·25-s − 1.43·26-s + 0.208·27-s − 2.71·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.06009116627\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.06009116627\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 4019 | \( 1+O(T) \) |
good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 3 | \( 1 + 2.35T + 3T^{2} \) |
| 5 | \( 1 - 1.72T + 5T^{2} \) |
| 7 | \( 1 + 3.13T + 7T^{2} \) |
| 11 | \( 1 + 4.23T + 11T^{2} \) |
| 13 | \( 1 - 2.84T + 13T^{2} \) |
| 17 | \( 1 + 1.50T + 17T^{2} \) |
| 19 | \( 1 + 1.85T + 19T^{2} \) |
| 23 | \( 1 + 7.40T + 23T^{2} \) |
| 29 | \( 1 + 1.18T + 29T^{2} \) |
| 31 | \( 1 + 7.01T + 31T^{2} \) |
| 37 | \( 1 + 0.469T + 37T^{2} \) |
| 41 | \( 1 - 3.60T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 + 6.26T + 53T^{2} \) |
| 59 | \( 1 + 8.36T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 0.827T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 - 5.98T + 97T^{2} \) |
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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
−8.458269686474775655253399205113, −7.87055619668900116381263185133, −6.86689399178207796576081707512, −6.34022419012498070294627474481, −5.91882392935690631118594309049, −5.13589484033568441858853570494, −3.60944491676434106848095704830, −2.44401650150032799168744475275, −1.61365628976074412748090402036, −0.19189277287159995031681582503,
0.19189277287159995031681582503, 1.61365628976074412748090402036, 2.44401650150032799168744475275, 3.60944491676434106848095704830, 5.13589484033568441858853570494, 5.91882392935690631118594309049, 6.34022419012498070294627474481, 6.86689399178207796576081707512, 7.87055619668900116381263185133, 8.458269686474775655253399205113