L(s) = 1 | − 2-s − 2.62·3-s + 4-s + 1.44·5-s + 2.62·6-s − 4.09·7-s − 8-s + 3.87·9-s − 1.44·10-s + 1.36·11-s − 2.62·12-s − 5.47·13-s + 4.09·14-s − 3.78·15-s + 16-s − 6.54·17-s − 3.87·18-s − 7.18·19-s + 1.44·20-s + 10.7·21-s − 1.36·22-s + 0.364·23-s + 2.62·24-s − 2.91·25-s + 5.47·26-s − 2.29·27-s − 4.09·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.51·3-s + 0.5·4-s + 0.646·5-s + 1.07·6-s − 1.54·7-s − 0.353·8-s + 1.29·9-s − 0.457·10-s + 0.410·11-s − 0.756·12-s − 1.51·13-s + 1.09·14-s − 0.978·15-s + 0.250·16-s − 1.58·17-s − 0.913·18-s − 1.64·19-s + 0.323·20-s + 2.34·21-s − 0.289·22-s + 0.0761·23-s + 0.535·24-s − 0.582·25-s + 1.07·26-s − 0.440·27-s − 0.773·28-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
\(L(1)\) |
\(\approx\) |
\(0.002780420286\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.002780420286\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.62T + 3T^{2} \) |
| 5 | \( 1 - 1.44T + 5T^{2} \) |
| 7 | \( 1 + 4.09T + 7T^{2} \) |
| 11 | \( 1 - 1.36T + 11T^{2} \) |
| 13 | \( 1 + 5.47T + 13T^{2} \) |
| 17 | \( 1 + 6.54T + 17T^{2} \) |
| 19 | \( 1 + 7.18T + 19T^{2} \) |
| 23 | \( 1 - 0.364T + 23T^{2} \) |
| 29 | \( 1 + 8.13T + 29T^{2} \) |
| 31 | \( 1 - 5.99T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 0.609T + 41T^{2} \) |
| 43 | \( 1 + 3.38T + 43T^{2} \) |
| 47 | \( 1 - 6.02T + 47T^{2} \) |
| 53 | \( 1 + 3.46T + 53T^{2} \) |
| 59 | \( 1 - 1.05T + 59T^{2} \) |
| 61 | \( 1 - 3.80T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 - 4.01T + 71T^{2} \) |
| 73 | \( 1 + 16.8T + 73T^{2} \) |
| 79 | \( 1 + 7.89T + 79T^{2} \) |
| 83 | \( 1 + 1.45T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 + 1.54T + 97T^{2} \) |
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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
−8.770025286394903622500770060948, −7.36738414578302691952251769210, −6.76899270140083916097604295519, −6.35154580822805640692079004509, −5.78956247394723299056674128676, −4.85326307250694503927897626718, −3.98398698672137100200439221674, −2.64104188120625107651657458364, −1.81970925045089792626097334470, −0.03386309186394984041411753085,
0.03386309186394984041411753085, 1.81970925045089792626097334470, 2.64104188120625107651657458364, 3.98398698672137100200439221674, 4.85326307250694503927897626718, 5.78956247394723299056674128676, 6.35154580822805640692079004509, 6.76899270140083916097604295519, 7.36738414578302691952251769210, 8.770025286394903622500770060948