| L(s) = 1 | + 1.14·3-s + 5-s + 0.357·7-s − 1.69·9-s + 3.73·11-s − 2.83·13-s + 1.14·15-s + 6.01·17-s − 6.73·19-s + 0.408·21-s + 3.51·23-s + 25-s − 5.36·27-s − 0.466·29-s + 7.02·31-s + 4.27·33-s + 0.357·35-s + 8.99·37-s − 3.24·39-s − 0.120·41-s + 8.01·43-s − 1.69·45-s − 6.49·47-s − 6.87·49-s + 6.88·51-s − 2.45·53-s + 3.73·55-s + ⋯ |
| L(s) = 1 | + 0.660·3-s + 0.447·5-s + 0.135·7-s − 0.563·9-s + 1.12·11-s − 0.785·13-s + 0.295·15-s + 1.45·17-s − 1.54·19-s + 0.0892·21-s + 0.733·23-s + 0.200·25-s − 1.03·27-s − 0.0866·29-s + 1.26·31-s + 0.744·33-s + 0.0604·35-s + 1.47·37-s − 0.519·39-s − 0.0188·41-s + 1.22·43-s − 0.252·45-s − 0.946·47-s − 0.981·49-s + 0.963·51-s − 0.337·53-s + 0.503·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.878705914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.878705914\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 + T \) |
| good | 3 | \( 1 - 1.14T + 3T^{2} \) |
| 7 | \( 1 - 0.357T + 7T^{2} \) |
| 11 | \( 1 - 3.73T + 11T^{2} \) |
| 13 | \( 1 + 2.83T + 13T^{2} \) |
| 17 | \( 1 - 6.01T + 17T^{2} \) |
| 19 | \( 1 + 6.73T + 19T^{2} \) |
| 23 | \( 1 - 3.51T + 23T^{2} \) |
| 29 | \( 1 + 0.466T + 29T^{2} \) |
| 31 | \( 1 - 7.02T + 31T^{2} \) |
| 37 | \( 1 - 8.99T + 37T^{2} \) |
| 41 | \( 1 + 0.120T + 41T^{2} \) |
| 43 | \( 1 - 8.01T + 43T^{2} \) |
| 47 | \( 1 + 6.49T + 47T^{2} \) |
| 53 | \( 1 + 2.45T + 53T^{2} \) |
| 59 | \( 1 - 3.14T + 59T^{2} \) |
| 61 | \( 1 + 0.565T + 61T^{2} \) |
| 67 | \( 1 - 3.57T + 67T^{2} \) |
| 71 | \( 1 + 0.0633T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 6.68T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 - 4.85T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−8.093837997869849392715065398400, −7.55471587855923333068446633155, −6.50189263671206091155724413268, −6.12890288259440120526563893196, −5.16225931612591524098021400393, −4.41065312015647598285843601303, −3.52312535781244486865176435592, −2.74943019150493496942014812703, −1.99756288189821115933140318707, −0.877071531784472227425680516335,
0.877071531784472227425680516335, 1.99756288189821115933140318707, 2.74943019150493496942014812703, 3.52312535781244486865176435592, 4.41065312015647598285843601303, 5.16225931612591524098021400393, 6.12890288259440120526563893196, 6.50189263671206091155724413268, 7.55471587855923333068446633155, 8.093837997869849392715065398400
