| L(s) = 1 | + 2.45·3-s − 1.72·5-s − 4.96·7-s + 3.02·9-s + 3.66·11-s − 4.13·13-s − 4.24·15-s + 17-s + 3.33·19-s − 12.1·21-s + 2.88·23-s − 2.01·25-s + 0.0685·27-s + 0.607·29-s − 1.49·31-s + 8.98·33-s + 8.58·35-s + 8.68·37-s − 10.1·39-s + 4.95·41-s + 6.72·43-s − 5.23·45-s + 0.874·47-s + 17.6·49-s + 2.45·51-s + 3.88·53-s − 6.33·55-s + ⋯ |
| L(s) = 1 | + 1.41·3-s − 0.773·5-s − 1.87·7-s + 1.00·9-s + 1.10·11-s − 1.14·13-s − 1.09·15-s + 0.242·17-s + 0.764·19-s − 2.65·21-s + 0.601·23-s − 0.402·25-s + 0.0131·27-s + 0.112·29-s − 0.268·31-s + 1.56·33-s + 1.45·35-s + 1.42·37-s − 1.62·39-s + 0.774·41-s + 1.02·43-s − 0.780·45-s + 0.127·47-s + 2.52·49-s + 0.343·51-s + 0.533·53-s − 0.853·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.085464035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.085464035\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 - 2.45T + 3T^{2} \) |
| 5 | \( 1 + 1.72T + 5T^{2} \) |
| 7 | \( 1 + 4.96T + 7T^{2} \) |
| 11 | \( 1 - 3.66T + 11T^{2} \) |
| 13 | \( 1 + 4.13T + 13T^{2} \) |
| 19 | \( 1 - 3.33T + 19T^{2} \) |
| 23 | \( 1 - 2.88T + 23T^{2} \) |
| 29 | \( 1 - 0.607T + 29T^{2} \) |
| 31 | \( 1 + 1.49T + 31T^{2} \) |
| 37 | \( 1 - 8.68T + 37T^{2} \) |
| 41 | \( 1 - 4.95T + 41T^{2} \) |
| 43 | \( 1 - 6.72T + 43T^{2} \) |
| 47 | \( 1 - 0.874T + 47T^{2} \) |
| 53 | \( 1 - 3.88T + 53T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 - 15.7T + 67T^{2} \) |
| 71 | \( 1 + 4.83T + 71T^{2} \) |
| 73 | \( 1 - 1.47T + 73T^{2} \) |
| 79 | \( 1 + 7.91T + 79T^{2} \) |
| 83 | \( 1 - 2.71T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−8.547967803494544101261989149066, −7.59170749589491410072503149504, −7.23651262268429891962675599049, −6.47745797490650143856395092948, −5.53788701605881281418301994514, −4.16423950732428628504963502650, −3.75267741351566231445099384698, −2.98828782878543678465586671610, −2.39863047946078044833942773123, −0.75096045680258998603152093728,
0.75096045680258998603152093728, 2.39863047946078044833942773123, 2.98828782878543678465586671610, 3.75267741351566231445099384698, 4.16423950732428628504963502650, 5.53788701605881281418301994514, 6.47745797490650143856395092948, 7.23651262268429891962675599049, 7.59170749589491410072503149504, 8.547967803494544101261989149066
