L(s) = 1 | − 2-s + 2.22·3-s + 4-s − 2.04·5-s − 2.22·6-s + 1.48·7-s − 8-s + 1.93·9-s + 2.04·10-s + 4.55·11-s + 2.22·12-s − 4.47·13-s − 1.48·14-s − 4.54·15-s + 16-s − 4.46·17-s − 1.93·18-s + 6.65·19-s − 2.04·20-s + 3.30·21-s − 4.55·22-s − 2.78·23-s − 2.22·24-s − 0.809·25-s + 4.47·26-s − 2.36·27-s + 1.48·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.28·3-s + 0.5·4-s − 0.915·5-s − 0.907·6-s + 0.561·7-s − 0.353·8-s + 0.645·9-s + 0.647·10-s + 1.37·11-s + 0.641·12-s − 1.23·13-s − 0.397·14-s − 1.17·15-s + 0.250·16-s − 1.08·17-s − 0.456·18-s + 1.52·19-s − 0.457·20-s + 0.720·21-s − 0.972·22-s − 0.580·23-s − 0.453·24-s − 0.161·25-s + 0.876·26-s − 0.454·27-s + 0.280·28-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
\(L(1)\) |
\(\approx\) |
\(1.949722174\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.949722174\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.22T + 3T^{2} \) |
| 5 | \( 1 + 2.04T + 5T^{2} \) |
| 7 | \( 1 - 1.48T + 7T^{2} \) |
| 11 | \( 1 - 4.55T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 + 4.46T + 17T^{2} \) |
| 19 | \( 1 - 6.65T + 19T^{2} \) |
| 23 | \( 1 + 2.78T + 23T^{2} \) |
| 29 | \( 1 + 1.98T + 29T^{2} \) |
| 31 | \( 1 - 5.60T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 + 6.43T + 41T^{2} \) |
| 43 | \( 1 - 5.49T + 43T^{2} \) |
| 47 | \( 1 - 13.6T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 5.64T + 61T^{2} \) |
| 67 | \( 1 + 0.320T + 67T^{2} \) |
| 71 | \( 1 + 6.67T + 71T^{2} \) |
| 73 | \( 1 - 7.34T + 73T^{2} \) |
| 79 | \( 1 - 17.2T + 79T^{2} \) |
| 83 | \( 1 - 3.97T + 83T^{2} \) |
| 89 | \( 1 + 8.65T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−8.331205284729505290999169791943, −7.87666086670009467657042222917, −7.32439434756437226989054460990, −6.64970575818440904484372590185, −5.46037852042880141799485820789, −4.27811097776112440430918893789, −3.84318783262141265640313223481, −2.75134387415706657432158689353, −2.08717736959978380495230312403, −0.839999849746655035498690978880,
0.839999849746655035498690978880, 2.08717736959978380495230312403, 2.75134387415706657432158689353, 3.84318783262141265640313223481, 4.27811097776112440430918893789, 5.46037852042880141799485820789, 6.64970575818440904484372590185, 7.32439434756437226989054460990, 7.87666086670009467657042222917, 8.331205284729505290999169791943