| L(s) = 1 | − 2.97·5-s − 2.27·7-s − 3.69·11-s + 3.41·13-s − 1.74·17-s − 7.76·19-s − 2.16·23-s + 3.82·25-s + 5.43·29-s − 8.70·31-s + 6.75·35-s + 0.585·37-s − 10.1·41-s − 1.33·43-s − 12.6·47-s − 1.82·49-s − 5.43·53-s + 10.9·55-s + 4.32·59-s + 7.41·61-s − 10.1·65-s + 6.43·67-s − 6.12·71-s + 10.4·73-s + 8.40·77-s + 2.27·79-s − 9.81·83-s + ⋯ |
| L(s) = 1 | − 1.32·5-s − 0.859·7-s − 1.11·11-s + 0.946·13-s − 0.422·17-s − 1.78·19-s − 0.451·23-s + 0.765·25-s + 1.00·29-s − 1.56·31-s + 1.14·35-s + 0.0963·37-s − 1.58·41-s − 0.203·43-s − 1.84·47-s − 0.261·49-s − 0.746·53-s + 1.48·55-s + 0.563·59-s + 0.949·61-s − 1.25·65-s + 0.785·67-s − 0.726·71-s + 1.22·73-s + 0.957·77-s + 0.255·79-s − 1.07·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1875464277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1875464277\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.97T + 5T^{2} \) |
| 7 | \( 1 + 2.27T + 7T^{2} \) |
| 11 | \( 1 + 3.69T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 + 1.74T + 17T^{2} \) |
| 19 | \( 1 + 7.76T + 19T^{2} \) |
| 23 | \( 1 + 2.16T + 23T^{2} \) |
| 29 | \( 1 - 5.43T + 29T^{2} \) |
| 31 | \( 1 + 8.70T + 31T^{2} \) |
| 37 | \( 1 - 0.585T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 1.33T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 + 5.43T + 53T^{2} \) |
| 59 | \( 1 - 4.32T + 59T^{2} \) |
| 61 | \( 1 - 7.41T + 61T^{2} \) |
| 67 | \( 1 - 6.43T + 67T^{2} \) |
| 71 | \( 1 + 6.12T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 2.27T + 79T^{2} \) |
| 83 | \( 1 + 9.81T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−7.970665125674490696589941810285, −6.76758832900697139563322206394, −6.69418343508139476608477816261, −5.67994458979045097552669637292, −4.85331439104281948240778956750, −4.06095371830436672848168304012, −3.55212088709656365255522962124, −2.79882165478119030720936055212, −1.76585111144122028862934721576, −0.19779492308396867995413960966,
0.19779492308396867995413960966, 1.76585111144122028862934721576, 2.79882165478119030720936055212, 3.55212088709656365255522962124, 4.06095371830436672848168304012, 4.85331439104281948240778956750, 5.67994458979045097552669637292, 6.69418343508139476608477816261, 6.76758832900697139563322206394, 7.970665125674490696589941810285
