| L(s) = 1 | + 1.47·2-s + 3-s + 0.185·4-s + 2.70·5-s + 1.47·6-s + 1.84·7-s − 2.68·8-s + 9-s + 3.99·10-s + 0.185·12-s − 5.40·13-s + 2.72·14-s + 2.70·15-s − 4.33·16-s + 6.21·17-s + 1.47·18-s + 19-s + 0.502·20-s + 1.84·21-s − 2.10·23-s − 2.68·24-s + 2.31·25-s − 7.99·26-s + 27-s + 0.342·28-s + 5.30·29-s + 3.99·30-s + ⋯ |
| L(s) = 1 | + 1.04·2-s + 0.577·3-s + 0.0928·4-s + 1.20·5-s + 0.603·6-s + 0.696·7-s − 0.948·8-s + 0.333·9-s + 1.26·10-s + 0.0535·12-s − 1.49·13-s + 0.728·14-s + 0.698·15-s − 1.08·16-s + 1.50·17-s + 0.348·18-s + 0.229·19-s + 0.112·20-s + 0.402·21-s − 0.439·23-s − 0.547·24-s + 0.463·25-s − 1.56·26-s + 0.192·27-s + 0.0646·28-s + 0.986·29-s + 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6897 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6897 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.339385858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.339385858\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.47T + 2T^{2} \) |
| 5 | \( 1 - 2.70T + 5T^{2} \) |
| 7 | \( 1 - 1.84T + 7T^{2} \) |
| 13 | \( 1 + 5.40T + 13T^{2} \) |
| 17 | \( 1 - 6.21T + 17T^{2} \) |
| 23 | \( 1 + 2.10T + 23T^{2} \) |
| 29 | \( 1 - 5.30T + 29T^{2} \) |
| 31 | \( 1 + 6.24T + 31T^{2} \) |
| 37 | \( 1 - 8.73T + 37T^{2} \) |
| 41 | \( 1 - 2.33T + 41T^{2} \) |
| 43 | \( 1 - 6.57T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 8.49T + 53T^{2} \) |
| 59 | \( 1 - 14.5T + 59T^{2} \) |
| 61 | \( 1 - 2.18T + 61T^{2} \) |
| 67 | \( 1 - 0.202T + 67T^{2} \) |
| 71 | \( 1 - 4.57T + 71T^{2} \) |
| 73 | \( 1 + 5.88T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 2.12T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 - 0.722T + 97T^{2} \) |
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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
−7.77567585767019164952245378020, −7.33413622925125640573451287295, −6.28609331744815719509966714805, −5.56394458374062489514053504455, −5.22588731521636847291046752661, −4.41562120352598554138366224844, −3.68227435509459315763356099163, −2.60050829429583518750715843041, −2.29993749591367554488671740684, −1.01619896373295207371953479790,
1.01619896373295207371953479790, 2.29993749591367554488671740684, 2.60050829429583518750715843041, 3.68227435509459315763356099163, 4.41562120352598554138366224844, 5.22588731521636847291046752661, 5.56394458374062489514053504455, 6.28609331744815719509966714805, 7.33413622925125640573451287295, 7.77567585767019164952245378020
