| L(s) = 1 | + 1.26·3-s + 3.95·5-s − 4.74·7-s − 1.39·9-s − 3.37·13-s + 5.00·15-s + 1.77·17-s − 19-s − 6.00·21-s + 6.66·23-s + 10.6·25-s − 5.56·27-s + 2.49·29-s + 3.70·31-s − 18.7·35-s − 4.97·37-s − 4.27·39-s − 8.19·41-s − 6.63·43-s − 5.50·45-s − 5.30·47-s + 15.4·49-s + 2.24·51-s − 2.97·53-s − 1.26·57-s − 11.0·59-s + 7.58·61-s + ⋯ |
| L(s) = 1 | + 0.731·3-s + 1.76·5-s − 1.79·7-s − 0.464·9-s − 0.935·13-s + 1.29·15-s + 0.430·17-s − 0.229·19-s − 1.31·21-s + 1.39·23-s + 2.12·25-s − 1.07·27-s + 0.463·29-s + 0.665·31-s − 3.16·35-s − 0.818·37-s − 0.684·39-s − 1.27·41-s − 1.01·43-s − 0.820·45-s − 0.773·47-s + 2.21·49-s + 0.314·51-s − 0.408·53-s − 0.167·57-s − 1.43·59-s + 0.971·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.26T + 3T^{2} \) |
| 5 | \( 1 - 3.95T + 5T^{2} \) |
| 7 | \( 1 + 4.74T + 7T^{2} \) |
| 13 | \( 1 + 3.37T + 13T^{2} \) |
| 17 | \( 1 - 1.77T + 17T^{2} \) |
| 23 | \( 1 - 6.66T + 23T^{2} \) |
| 29 | \( 1 - 2.49T + 29T^{2} \) |
| 31 | \( 1 - 3.70T + 31T^{2} \) |
| 37 | \( 1 + 4.97T + 37T^{2} \) |
| 41 | \( 1 + 8.19T + 41T^{2} \) |
| 43 | \( 1 + 6.63T + 43T^{2} \) |
| 47 | \( 1 + 5.30T + 47T^{2} \) |
| 53 | \( 1 + 2.97T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 - 7.58T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 + 7.43T + 71T^{2} \) |
| 73 | \( 1 - 8.29T + 73T^{2} \) |
| 79 | \( 1 - 3.05T + 79T^{2} \) |
| 83 | \( 1 + 5.54T + 83T^{2} \) |
| 89 | \( 1 + 4.63T + 89T^{2} \) |
| 97 | \( 1 + 8.56T + 97T^{2} \) |
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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
−7.10711496285885395741969957688, −6.69454097533204159076571549113, −6.08014638795881072872200755938, −5.42626291409987921733228821676, −4.76377188598109535555433465975, −3.36907855509977890926032270579, −2.98532835048811683320691246860, −2.42958293236413863774730566810, −1.45236721096771900704507526463, 0,
1.45236721096771900704507526463, 2.42958293236413863774730566810, 2.98532835048811683320691246860, 3.36907855509977890926032270579, 4.76377188598109535555433465975, 5.42626291409987921733228821676, 6.08014638795881072872200755938, 6.69454097533204159076571549113, 7.10711496285885395741969957688
