| L(s) = 1 | + 2.68·2-s + 1.11·3-s + 5.20·4-s − 2.50·5-s + 2.99·6-s + 8.59·8-s − 1.75·9-s − 6.71·10-s + 3.90·11-s + 5.80·12-s + 3.81·13-s − 2.79·15-s + 12.6·16-s + 0.499·17-s − 4.70·18-s + 19-s − 13.0·20-s + 10.4·22-s − 6.75·23-s + 9.59·24-s + 1.25·25-s + 10.2·26-s − 5.30·27-s + 1.90·29-s − 7.49·30-s − 4.15·31-s + 16.7·32-s + ⋯ |
| L(s) = 1 | + 1.89·2-s + 0.644·3-s + 2.60·4-s − 1.11·5-s + 1.22·6-s + 3.03·8-s − 0.584·9-s − 2.12·10-s + 1.17·11-s + 1.67·12-s + 1.05·13-s − 0.721·15-s + 3.16·16-s + 0.121·17-s − 1.10·18-s + 0.229·19-s − 2.90·20-s + 2.23·22-s − 1.40·23-s + 1.95·24-s + 0.250·25-s + 2.00·26-s − 1.02·27-s + 0.354·29-s − 1.36·30-s − 0.745·31-s + 2.96·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.284172105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.284172105\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.68T + 2T^{2} \) |
| 3 | \( 1 - 1.11T + 3T^{2} \) |
| 5 | \( 1 + 2.50T + 5T^{2} \) |
| 11 | \( 1 - 3.90T + 11T^{2} \) |
| 13 | \( 1 - 3.81T + 13T^{2} \) |
| 17 | \( 1 - 0.499T + 17T^{2} \) |
| 23 | \( 1 + 6.75T + 23T^{2} \) |
| 29 | \( 1 - 1.90T + 29T^{2} \) |
| 31 | \( 1 + 4.15T + 31T^{2} \) |
| 37 | \( 1 + 3.45T + 37T^{2} \) |
| 41 | \( 1 + 5.36T + 41T^{2} \) |
| 43 | \( 1 + 5.67T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 0.368T + 53T^{2} \) |
| 59 | \( 1 + 3.72T + 59T^{2} \) |
| 61 | \( 1 + 9.33T + 61T^{2} \) |
| 67 | \( 1 + 8.24T + 67T^{2} \) |
| 71 | \( 1 - 0.281T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 6.71T + 83T^{2} \) |
| 89 | \( 1 - 6.27T + 89T^{2} \) |
| 97 | \( 1 + 0.490T + 97T^{2} \) |
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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
−10.49656844675585511405238828894, −9.033122148166502753389320742305, −8.131483052434592085056101128485, −7.36408179985708926631947555861, −6.37246073556090937658271404991, −5.68385004507807532686329733480, −4.40721342984398416351842854816, −3.70855216487150237984231758415, −3.24435526331884126104964809818, −1.82109391442292736305464424397,
1.82109391442292736305464424397, 3.24435526331884126104964809818, 3.70855216487150237984231758415, 4.40721342984398416351842854816, 5.68385004507807532686329733480, 6.37246073556090937658271404991, 7.36408179985708926631947555861, 8.131483052434592085056101128485, 9.033122148166502753389320742305, 10.49656844675585511405238828894
