L(s) = 1 | − 2.28·2-s + 1.07·3-s + 3.21·4-s − 3.29·5-s − 2.45·6-s − 2.40·7-s − 2.77·8-s − 1.84·9-s + 7.52·10-s − 5.96·11-s + 3.45·12-s − 3.44·13-s + 5.49·14-s − 3.53·15-s − 0.0899·16-s + 0.944·17-s + 4.21·18-s + 5.65·19-s − 10.5·20-s − 2.58·21-s + 13.6·22-s − 0.394·23-s − 2.98·24-s + 5.84·25-s + 7.86·26-s − 5.20·27-s − 7.73·28-s + ⋯ |
L(s) = 1 | − 1.61·2-s + 0.620·3-s + 1.60·4-s − 1.47·5-s − 1.00·6-s − 0.908·7-s − 0.981·8-s − 0.615·9-s + 2.37·10-s − 1.79·11-s + 0.996·12-s − 0.955·13-s + 1.46·14-s − 0.913·15-s − 0.0224·16-s + 0.228·17-s + 0.994·18-s + 1.29·19-s − 2.36·20-s − 0.563·21-s + 2.90·22-s − 0.0823·23-s − 0.608·24-s + 1.16·25-s + 1.54·26-s − 1.00·27-s − 1.46·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2587925220\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2587925220\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 983 | \( 1 - T \) |
good | 2 | \( 1 + 2.28T + 2T^{2} \) |
| 3 | \( 1 - 1.07T + 3T^{2} \) |
| 5 | \( 1 + 3.29T + 5T^{2} \) |
| 7 | \( 1 + 2.40T + 7T^{2} \) |
| 11 | \( 1 + 5.96T + 11T^{2} \) |
| 13 | \( 1 + 3.44T + 13T^{2} \) |
| 17 | \( 1 - 0.944T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 0.394T + 23T^{2} \) |
| 29 | \( 1 + 4.14T + 29T^{2} \) |
| 31 | \( 1 - 8.53T + 31T^{2} \) |
| 37 | \( 1 - 9.49T + 37T^{2} \) |
| 41 | \( 1 - 6.72T + 41T^{2} \) |
| 43 | \( 1 - 0.522T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 + 0.326T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 + 7.78T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 1.53T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 2.49T + 97T^{2} \) |
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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
−9.751218176102875915902869529131, −9.203852341592989914036775845274, −8.171100401063264959218095908134, −7.63986278450778050262399370346, −7.44748071098611155352306356335, −5.97128737631753408361606178715, −4.61826505106930159944656133679, −3.10827525314556065715330920443, −2.63157007270314257401159566193, −0.45408719764857708169035078788,
0.45408719764857708169035078788, 2.63157007270314257401159566193, 3.10827525314556065715330920443, 4.61826505106930159944656133679, 5.97128737631753408361606178715, 7.44748071098611155352306356335, 7.63986278450778050262399370346, 8.171100401063264959218095908134, 9.203852341592989914036775845274, 9.751218176102875915902869529131