| L(s) = 1 | − 1.79·3-s + 3.50·5-s − 0.885·7-s + 0.219·9-s − 2.51·11-s + 3.89·13-s − 6.28·15-s + 2.93·17-s − 1.92·19-s + 1.58·21-s + 3.51·23-s + 7.26·25-s + 4.98·27-s + 1.32·29-s + 4.81·31-s + 4.51·33-s − 3.10·35-s − 7.25·37-s − 6.98·39-s − 1.70·41-s + 2.90·43-s + 0.770·45-s − 1.51·47-s − 6.21·49-s − 5.27·51-s − 2.91·53-s − 8.82·55-s + ⋯ |
| L(s) = 1 | − 1.03·3-s + 1.56·5-s − 0.334·7-s + 0.0733·9-s − 0.759·11-s + 1.08·13-s − 1.62·15-s + 0.712·17-s − 0.442·19-s + 0.346·21-s + 0.732·23-s + 1.45·25-s + 0.960·27-s + 0.246·29-s + 0.864·31-s + 0.786·33-s − 0.524·35-s − 1.19·37-s − 1.11·39-s − 0.266·41-s + 0.442·43-s + 0.114·45-s − 0.220·47-s − 0.887·49-s − 0.738·51-s − 0.400·53-s − 1.18·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.462578440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.462578440\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 3 | \( 1 + 1.79T + 3T^{2} \) |
| 5 | \( 1 - 3.50T + 5T^{2} \) |
| 7 | \( 1 + 0.885T + 7T^{2} \) |
| 11 | \( 1 + 2.51T + 11T^{2} \) |
| 13 | \( 1 - 3.89T + 13T^{2} \) |
| 17 | \( 1 - 2.93T + 17T^{2} \) |
| 19 | \( 1 + 1.92T + 19T^{2} \) |
| 23 | \( 1 - 3.51T + 23T^{2} \) |
| 29 | \( 1 - 1.32T + 29T^{2} \) |
| 31 | \( 1 - 4.81T + 31T^{2} \) |
| 37 | \( 1 + 7.25T + 37T^{2} \) |
| 41 | \( 1 + 1.70T + 41T^{2} \) |
| 43 | \( 1 - 2.90T + 43T^{2} \) |
| 47 | \( 1 + 1.51T + 47T^{2} \) |
| 53 | \( 1 + 2.91T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 - 9.21T + 61T^{2} \) |
| 67 | \( 1 - 5.89T + 67T^{2} \) |
| 71 | \( 1 + 1.68T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 - 8.31T + 97T^{2} \) |
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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
−9.824427338029682123883418536461, −8.924674785215582207901030944894, −8.101327386229308210168121621231, −6.66201072873368996608538326217, −6.31552390730657489548481482748, −5.43405288980609720575085755658, −5.02629306257946132745493115276, −3.41149587006062697164625348846, −2.27705267046501090372870094314, −0.957666243795140857488366111043,
0.957666243795140857488366111043, 2.27705267046501090372870094314, 3.41149587006062697164625348846, 5.02629306257946132745493115276, 5.43405288980609720575085755658, 6.31552390730657489548481482748, 6.66201072873368996608538326217, 8.101327386229308210168121621231, 8.924674785215582207901030944894, 9.824427338029682123883418536461
