| L(s) = 1 | − 2.66·2-s + 0.662·3-s + 5.12·4-s − 2.84·5-s − 1.76·6-s − 0.180·7-s − 8.33·8-s − 2.56·9-s + 7.58·10-s − 11-s + 3.39·12-s + 1.26·13-s + 0.480·14-s − 1.88·15-s + 12.0·16-s + 1.06·17-s + 6.83·18-s + 4.26·19-s − 14.5·20-s − 0.119·21-s + 2.66·22-s + 9.55·23-s − 5.52·24-s + 3.08·25-s − 3.37·26-s − 3.68·27-s − 0.923·28-s + ⋯ |
| L(s) = 1 | − 1.88·2-s + 0.382·3-s + 2.56·4-s − 1.27·5-s − 0.721·6-s − 0.0681·7-s − 2.94·8-s − 0.853·9-s + 2.40·10-s − 0.301·11-s + 0.979·12-s + 0.351·13-s + 0.128·14-s − 0.486·15-s + 3.00·16-s + 0.259·17-s + 1.61·18-s + 0.977·19-s − 3.25·20-s − 0.0260·21-s + 0.569·22-s + 1.99·23-s − 1.12·24-s + 0.617·25-s − 0.662·26-s − 0.709·27-s − 0.174·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 + 2.66T + 2T^{2} \) |
| 3 | \( 1 - 0.662T + 3T^{2} \) |
| 5 | \( 1 + 2.84T + 5T^{2} \) |
| 7 | \( 1 + 0.180T + 7T^{2} \) |
| 13 | \( 1 - 1.26T + 13T^{2} \) |
| 17 | \( 1 - 1.06T + 17T^{2} \) |
| 19 | \( 1 - 4.26T + 19T^{2} \) |
| 23 | \( 1 - 9.55T + 23T^{2} \) |
| 29 | \( 1 - 2.29T + 29T^{2} \) |
| 31 | \( 1 - 5.97T + 31T^{2} \) |
| 37 | \( 1 + 9.40T + 37T^{2} \) |
| 41 | \( 1 + 9.00T + 41T^{2} \) |
| 43 | \( 1 - 8.24T + 43T^{2} \) |
| 47 | \( 1 - 0.606T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 3.71T + 61T^{2} \) |
| 67 | \( 1 + 2.18T + 67T^{2} \) |
| 71 | \( 1 + 9.13T + 71T^{2} \) |
| 73 | \( 1 - 0.113T + 73T^{2} \) |
| 79 | \( 1 + 9.53T + 79T^{2} \) |
| 83 | \( 1 - 7.25T + 83T^{2} \) |
| 89 | \( 1 + 7.31T + 89T^{2} \) |
| 97 | \( 1 - 0.678T + 97T^{2} \) |
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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
−8.884783363132954108767100425587, −8.463430198514850599112618620983, −7.74130591405612970802529018928, −7.20939359193917785055077572603, −6.28900089714487399620361546844, −5.03960686063303205098034728777, −3.33538117513081054483150762461, −2.87258691131999248791011811911, −1.26672769494788302003924607420, 0,
1.26672769494788302003924607420, 2.87258691131999248791011811911, 3.33538117513081054483150762461, 5.03960686063303205098034728777, 6.28900089714487399620361546844, 7.20939359193917785055077572603, 7.74130591405612970802529018928, 8.463430198514850599112618620983, 8.884783363132954108767100425587
