| L(s) = 1 | − 2.73·5-s + 3.46·7-s − 4.19·11-s + 4.46·13-s + 2.73·17-s − 2.46·19-s + 5.26·23-s + 2.46·25-s − 9.46·29-s + 4.46·31-s − 9.46·35-s + 9.46·37-s − 2.53·41-s − 7.92·43-s + 6.92·47-s + 4.99·49-s − 9.66·53-s + 11.4·55-s + 12.1·59-s − 8.46·61-s − 12.1·65-s + 10.4·67-s + 4.19·71-s − 9.39·73-s − 14.5·77-s − 0.0717·79-s + 9.26·83-s + ⋯ |
| L(s) = 1 | − 1.22·5-s + 1.30·7-s − 1.26·11-s + 1.23·13-s + 0.662·17-s − 0.565·19-s + 1.09·23-s + 0.492·25-s − 1.75·29-s + 0.801·31-s − 1.59·35-s + 1.55·37-s − 0.396·41-s − 1.20·43-s + 1.01·47-s + 0.714·49-s − 1.32·53-s + 1.54·55-s + 1.58·59-s − 1.08·61-s − 1.51·65-s + 1.27·67-s + 0.497·71-s − 1.09·73-s − 1.65·77-s − 0.00807·79-s + 1.01·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.622307420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.622307420\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.73T + 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 + 4.19T + 11T^{2} \) |
| 13 | \( 1 - 4.46T + 13T^{2} \) |
| 17 | \( 1 - 2.73T + 17T^{2} \) |
| 19 | \( 1 + 2.46T + 19T^{2} \) |
| 23 | \( 1 - 5.26T + 23T^{2} \) |
| 29 | \( 1 + 9.46T + 29T^{2} \) |
| 31 | \( 1 - 4.46T + 31T^{2} \) |
| 37 | \( 1 - 9.46T + 37T^{2} \) |
| 41 | \( 1 + 2.53T + 41T^{2} \) |
| 43 | \( 1 + 7.92T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 9.66T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 + 8.46T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 4.19T + 71T^{2} \) |
| 73 | \( 1 + 9.39T + 73T^{2} \) |
| 79 | \( 1 + 0.0717T + 79T^{2} \) |
| 83 | \( 1 - 9.26T + 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + 1.92T + 97T^{2} \) |
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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
−8.267131419119923825539633601135, −7.83766939069795636329282997748, −7.34957648677080853615715551467, −6.21991331105646937348077517403, −5.32686538633766179813003651526, −4.70271365295861857155172037684, −3.89231394326945207726135400753, −3.10952959587601461314267057056, −1.91863074373960696969527492027, −0.74782880774744345182646019146,
0.74782880774744345182646019146, 1.91863074373960696969527492027, 3.10952959587601461314267057056, 3.89231394326945207726135400753, 4.70271365295861857155172037684, 5.32686538633766179813003651526, 6.21991331105646937348077517403, 7.34957648677080853615715551467, 7.83766939069795636329282997748, 8.267131419119923825539633601135
