L(s) = 1 | − 2-s + 2.41·3-s + 4-s − 2.81·5-s − 2.41·6-s − 1.01·7-s − 8-s + 2.81·9-s + 2.81·10-s − 1.71·11-s + 2.41·12-s + 6.89·13-s + 1.01·14-s − 6.79·15-s + 16-s − 0.953·17-s − 2.81·18-s − 0.313·19-s − 2.81·20-s − 2.44·21-s + 1.71·22-s − 0.129·23-s − 2.41·24-s + 2.93·25-s − 6.89·26-s − 0.450·27-s − 1.01·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.39·3-s + 0.5·4-s − 1.26·5-s − 0.984·6-s − 0.383·7-s − 0.353·8-s + 0.937·9-s + 0.890·10-s − 0.516·11-s + 0.696·12-s + 1.91·13-s + 0.271·14-s − 1.75·15-s + 0.250·16-s − 0.231·17-s − 0.663·18-s − 0.0720·19-s − 0.630·20-s − 0.534·21-s + 0.365·22-s − 0.0270·23-s − 0.492·24-s + 0.587·25-s − 1.35·26-s − 0.0866·27-s − 0.191·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.630319949\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.630319949\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
good | 3 | \( 1 - 2.41T + 3T^{2} \) |
| 5 | \( 1 + 2.81T + 5T^{2} \) |
| 7 | \( 1 + 1.01T + 7T^{2} \) |
| 11 | \( 1 + 1.71T + 11T^{2} \) |
| 13 | \( 1 - 6.89T + 13T^{2} \) |
| 17 | \( 1 + 0.953T + 17T^{2} \) |
| 19 | \( 1 + 0.313T + 19T^{2} \) |
| 23 | \( 1 + 0.129T + 23T^{2} \) |
| 29 | \( 1 + 0.276T + 29T^{2} \) |
| 31 | \( 1 + 5.01T + 31T^{2} \) |
| 37 | \( 1 - 2.43T + 37T^{2} \) |
| 41 | \( 1 - 8.42T + 41T^{2} \) |
| 43 | \( 1 - 7.86T + 43T^{2} \) |
| 47 | \( 1 - 0.415T + 47T^{2} \) |
| 53 | \( 1 - 3.62T + 53T^{2} \) |
| 59 | \( 1 - 0.0527T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 - 7.46T + 67T^{2} \) |
| 71 | \( 1 + 7.28T + 71T^{2} \) |
| 73 | \( 1 - 9.97T + 73T^{2} \) |
| 79 | \( 1 + 7.59T + 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 + 7.70T + 97T^{2} \) |
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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
−8.468293720816651198388539411580, −7.86910978245239614362942823586, −7.42046025605801092097544492364, −6.50456611555660476341617058979, −5.62866998089792002457559067789, −4.15875900984680031022361001344, −3.69590794679020448826432282894, −2.99950382094285710691783631500, −2.05148264122823437493485904102, −0.75237164464824595725658754723,
0.75237164464824595725658754723, 2.05148264122823437493485904102, 2.99950382094285710691783631500, 3.69590794679020448826432282894, 4.15875900984680031022361001344, 5.62866998089792002457559067789, 6.50456611555660476341617058979, 7.42046025605801092097544492364, 7.86910978245239614362942823586, 8.468293720816651198388539411580