| L(s) = 1 | − 2-s + 2.62·3-s + 4-s + 2.00·5-s − 2.62·6-s + 0.373·7-s − 8-s + 3.90·9-s − 2.00·10-s − 1.73·11-s + 2.62·12-s − 1.83·13-s − 0.373·14-s + 5.27·15-s + 16-s − 1.21·17-s − 3.90·18-s + 7.37·19-s + 2.00·20-s + 0.982·21-s + 1.73·22-s + 8.08·23-s − 2.62·24-s − 0.964·25-s + 1.83·26-s + 2.38·27-s + 0.373·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.51·3-s + 0.5·4-s + 0.898·5-s − 1.07·6-s + 0.141·7-s − 0.353·8-s + 1.30·9-s − 0.635·10-s − 0.521·11-s + 0.758·12-s − 0.509·13-s − 0.0998·14-s + 1.36·15-s + 0.250·16-s − 0.294·17-s − 0.920·18-s + 1.69·19-s + 0.449·20-s + 0.214·21-s + 0.368·22-s + 1.68·23-s − 0.536·24-s − 0.192·25-s + 0.360·26-s + 0.458·27-s + 0.0706·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.213898260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.213898260\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 - 2.62T + 3T^{2} \) |
| 5 | \( 1 - 2.00T + 5T^{2} \) |
| 7 | \( 1 - 0.373T + 7T^{2} \) |
| 11 | \( 1 + 1.73T + 11T^{2} \) |
| 13 | \( 1 + 1.83T + 13T^{2} \) |
| 17 | \( 1 + 1.21T + 17T^{2} \) |
| 19 | \( 1 - 7.37T + 19T^{2} \) |
| 23 | \( 1 - 8.08T + 23T^{2} \) |
| 29 | \( 1 - 7.03T + 29T^{2} \) |
| 31 | \( 1 + 8.79T + 31T^{2} \) |
| 37 | \( 1 + 2.45T + 37T^{2} \) |
| 41 | \( 1 - 2.74T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 + 2.86T + 47T^{2} \) |
| 53 | \( 1 + 2.22T + 53T^{2} \) |
| 59 | \( 1 + 7.28T + 59T^{2} \) |
| 61 | \( 1 - 3.58T + 61T^{2} \) |
| 67 | \( 1 + 8.67T + 67T^{2} \) |
| 71 | \( 1 + 9.48T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 + 4.24T + 83T^{2} \) |
| 89 | \( 1 - 17.4T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−9.854246965441506830803815933455, −9.197747254150742202007951592306, −8.771025305030022562433349747209, −7.57881940742678820891137873325, −7.29919410717147803557767005143, −5.89444662565118889181734491441, −4.82618173131407098069204345582, −3.22785802056232158671060587440, −2.57731636585566383061267853511, −1.46832194973399051293646783991,
1.46832194973399051293646783991, 2.57731636585566383061267853511, 3.22785802056232158671060587440, 4.82618173131407098069204345582, 5.89444662565118889181734491441, 7.29919410717147803557767005143, 7.57881940742678820891137873325, 8.771025305030022562433349747209, 9.197747254150742202007951592306, 9.854246965441506830803815933455
