| L(s) = 1 | − 1.23·5-s − 3.23·7-s − 5.23·11-s + 3·13-s − 0.763·17-s + 2·19-s + 23-s − 3.47·25-s + 3·29-s − 6.70·31-s + 4.00·35-s − 1.23·37-s + 3.47·41-s − 2.23·47-s + 3.47·49-s − 0.472·53-s + 6.47·55-s + 6.47·59-s − 6.94·61-s − 3.70·65-s + 2.76·67-s + 12.2·71-s + 6.52·73-s + 16.9·77-s + 10.9·79-s − 8.76·83-s + 0.944·85-s + ⋯ |
| L(s) = 1 | − 0.552·5-s − 1.22·7-s − 1.57·11-s + 0.832·13-s − 0.185·17-s + 0.458·19-s + 0.208·23-s − 0.694·25-s + 0.557·29-s − 1.20·31-s + 0.676·35-s − 0.203·37-s + 0.542·41-s − 0.326·47-s + 0.496·49-s − 0.0648·53-s + 0.872·55-s + 0.842·59-s − 0.889·61-s − 0.459·65-s + 0.337·67-s + 1.45·71-s + 0.764·73-s + 1.93·77-s + 1.23·79-s − 0.961·83-s + 0.102·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9034607919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9034607919\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 + 3.23T + 7T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 + 0.763T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 6.70T + 31T^{2} \) |
| 37 | \( 1 + 1.23T + 37T^{2} \) |
| 41 | \( 1 - 3.47T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 2.23T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 - 6.47T + 59T^{2} \) |
| 61 | \( 1 + 6.94T + 61T^{2} \) |
| 67 | \( 1 - 2.76T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 - 6.52T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 8.76T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−8.562467977277851021976711926244, −7.85746500409114211897838590591, −7.22936716899332769411195005035, −6.36540720768802185934546343353, −5.65930968752196584556321377407, −4.84258485959937503398528219859, −3.71478035413279998549717650532, −3.20296792508710541779652772849, −2.18941875851915509258268960926, −0.53904469614056012023268066169,
0.53904469614056012023268066169, 2.18941875851915509258268960926, 3.20296792508710541779652772849, 3.71478035413279998549717650532, 4.84258485959937503398528219859, 5.65930968752196584556321377407, 6.36540720768802185934546343353, 7.22936716899332769411195005035, 7.85746500409114211897838590591, 8.562467977277851021976711926244
