| L(s) = 1 | − 1.43·3-s + 2.34·5-s + 7-s − 0.938·9-s − 5.38·11-s − 5.32·13-s − 3.36·15-s + 7.88·17-s − 6.06·19-s − 1.43·21-s + 23-s + 0.486·25-s + 5.65·27-s + 1.87·29-s + 5.83·31-s + 7.73·33-s + 2.34·35-s + 1.19·37-s + 7.64·39-s + 6.45·41-s + 6.93·43-s − 2.19·45-s + 10.8·47-s + 49-s − 11.3·51-s + 7.94·53-s − 12.6·55-s + ⋯ |
| L(s) = 1 | − 0.828·3-s + 1.04·5-s + 0.377·7-s − 0.312·9-s − 1.62·11-s − 1.47·13-s − 0.868·15-s + 1.91·17-s − 1.39·19-s − 0.313·21-s + 0.208·23-s + 0.0973·25-s + 1.08·27-s + 0.348·29-s + 1.04·31-s + 1.34·33-s + 0.395·35-s + 0.195·37-s + 1.22·39-s + 1.00·41-s + 1.05·43-s − 0.327·45-s + 1.58·47-s + 0.142·49-s − 1.58·51-s + 1.09·53-s − 1.70·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.271145470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.271145470\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 1.43T + 3T^{2} \) |
| 5 | \( 1 - 2.34T + 5T^{2} \) |
| 11 | \( 1 + 5.38T + 11T^{2} \) |
| 13 | \( 1 + 5.32T + 13T^{2} \) |
| 17 | \( 1 - 7.88T + 17T^{2} \) |
| 19 | \( 1 + 6.06T + 19T^{2} \) |
| 29 | \( 1 - 1.87T + 29T^{2} \) |
| 31 | \( 1 - 5.83T + 31T^{2} \) |
| 37 | \( 1 - 1.19T + 37T^{2} \) |
| 41 | \( 1 - 6.45T + 41T^{2} \) |
| 43 | \( 1 - 6.93T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 7.94T + 53T^{2} \) |
| 59 | \( 1 - 2.69T + 59T^{2} \) |
| 61 | \( 1 + 3.27T + 61T^{2} \) |
| 67 | \( 1 - 1.82T + 67T^{2} \) |
| 71 | \( 1 + 7.30T + 71T^{2} \) |
| 73 | \( 1 + 7.51T + 73T^{2} \) |
| 79 | \( 1 - 2.49T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−8.952124861270309423091234249328, −7.946051277824447043220527651145, −7.47745845393747487946949821058, −6.32698364145592302483833966044, −5.60663274255447663075438098491, −5.27364307912856941428784110370, −4.43070490509366344083924270889, −2.77868096168644635702955405365, −2.29550349600340777390301809616, −0.71768369482283642903654283793,
0.71768369482283642903654283793, 2.29550349600340777390301809616, 2.77868096168644635702955405365, 4.43070490509366344083924270889, 5.27364307912856941428784110370, 5.60663274255447663075438098491, 6.32698364145592302483833966044, 7.47745845393747487946949821058, 7.946051277824447043220527651145, 8.952124861270309423091234249328
