| L(s) = 1 | + 0.444·2-s − 1.68·3-s − 1.80·4-s + 3.41·5-s − 0.749·6-s + 1.25·7-s − 1.69·8-s − 0.157·9-s + 1.51·10-s + 3.03·12-s − 4.74·13-s + 0.558·14-s − 5.76·15-s + 2.85·16-s + 6.97·17-s − 0.0697·18-s − 5.14·19-s − 6.15·20-s − 2.11·21-s − 6.04·23-s + 2.84·24-s + 6.67·25-s − 2.10·26-s + 5.32·27-s − 2.26·28-s + 1.09·29-s − 2.56·30-s + ⋯ |
| L(s) = 1 | + 0.314·2-s − 0.973·3-s − 0.901·4-s + 1.52·5-s − 0.305·6-s + 0.474·7-s − 0.597·8-s − 0.0523·9-s + 0.480·10-s + 0.877·12-s − 1.31·13-s + 0.149·14-s − 1.48·15-s + 0.713·16-s + 1.69·17-s − 0.0164·18-s − 1.17·19-s − 1.37·20-s − 0.462·21-s − 1.26·23-s + 0.581·24-s + 1.33·25-s − 0.413·26-s + 1.02·27-s − 0.427·28-s + 0.203·29-s − 0.467·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.363316784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.363316784\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 0.444T + 2T^{2} \) |
| 3 | \( 1 + 1.68T + 3T^{2} \) |
| 5 | \( 1 - 3.41T + 5T^{2} \) |
| 7 | \( 1 - 1.25T + 7T^{2} \) |
| 13 | \( 1 + 4.74T + 13T^{2} \) |
| 17 | \( 1 - 6.97T + 17T^{2} \) |
| 19 | \( 1 + 5.14T + 19T^{2} \) |
| 23 | \( 1 + 6.04T + 23T^{2} \) |
| 29 | \( 1 - 1.09T + 29T^{2} \) |
| 31 | \( 1 - 2.37T + 31T^{2} \) |
| 37 | \( 1 + 1.69T + 37T^{2} \) |
| 41 | \( 1 + 3.79T + 41T^{2} \) |
| 43 | \( 1 + 4.85T + 43T^{2} \) |
| 47 | \( 1 + 6.96T + 47T^{2} \) |
| 53 | \( 1 - 6.62T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 - 8.58T + 73T^{2} \) |
| 79 | \( 1 - 0.574T + 79T^{2} \) |
| 83 | \( 1 - 9.44T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 18.0T + 97T^{2} \) |
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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
−8.194723168251229292957624748698, −6.89463526248742011619566465837, −6.26966724227895206210641084540, −5.57291697464612869940308234188, −5.23437114406676358910136030844, −4.72409062330902142521431456086, −3.68759700042676281131325792067, −2.61285206607579684530540472178, −1.75979084172846098638658696799, −0.59165569452944146884602098455,
0.59165569452944146884602098455, 1.75979084172846098638658696799, 2.61285206607579684530540472178, 3.68759700042676281131325792067, 4.72409062330902142521431456086, 5.23437114406676358910136030844, 5.57291697464612869940308234188, 6.26966724227895206210641084540, 6.89463526248742011619566465837, 8.194723168251229292957624748698
