L(s) = 1 | − 5-s + 3.44·7-s + 1.44·11-s + 5.89·13-s + 4.89·17-s + 4·19-s − 5.44·23-s − 4·25-s + 0.101·29-s + 2.55·31-s − 3.44·35-s − 0.898·37-s − 11.8·41-s − 2.34·43-s − 6.34·47-s + 4.89·49-s + 8.89·53-s − 1.44·55-s + 14.3·59-s − 7.89·61-s − 5.89·65-s + 12.3·67-s + 7.79·71-s − 4.89·73-s + 5·77-s + 13.4·79-s − 0.550·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.30·7-s + 0.437·11-s + 1.63·13-s + 1.18·17-s + 0.917·19-s − 1.13·23-s − 0.800·25-s + 0.0187·29-s + 0.458·31-s − 0.583·35-s − 0.147·37-s − 1.85·41-s − 0.358·43-s − 0.926·47-s + 0.699·49-s + 1.22·53-s − 0.195·55-s + 1.86·59-s − 1.01·61-s − 0.731·65-s + 1.50·67-s + 0.925·71-s − 0.573·73-s + 0.569·77-s + 1.51·79-s − 0.0604·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.283165327\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.283165327\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 - 3.44T + 7T^{2} \) |
| 11 | \( 1 - 1.44T + 11T^{2} \) |
| 13 | \( 1 - 5.89T + 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 5.44T + 23T^{2} \) |
| 29 | \( 1 - 0.101T + 29T^{2} \) |
| 31 | \( 1 - 2.55T + 31T^{2} \) |
| 37 | \( 1 + 0.898T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 2.34T + 43T^{2} \) |
| 47 | \( 1 + 6.34T + 47T^{2} \) |
| 53 | \( 1 - 8.89T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 + 7.89T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 - 7.79T + 71T^{2} \) |
| 73 | \( 1 + 4.89T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 0.550T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 + 3.89T + 97T^{2} \) |
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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
−8.516156387633717483264709018005, −8.244064769065772644651291537048, −7.56449533582697760679046666174, −6.57674458880576720026293652616, −5.68999466352124169823931246445, −5.02098153318161133497500135711, −3.91829561558959665979482764669, −3.45415152338218935489433914071, −1.88202084675789651931961824886, −1.05452782238790190390759779110,
1.05452782238790190390759779110, 1.88202084675789651931961824886, 3.45415152338218935489433914071, 3.91829561558959665979482764669, 5.02098153318161133497500135711, 5.68999466352124169823931246445, 6.57674458880576720026293652616, 7.56449533582697760679046666174, 8.244064769065772644651291537048, 8.516156387633717483264709018005