| L(s) = 1 | − 2.18·3-s + 4.04·5-s + 7-s + 1.76·9-s + 3.98·11-s + 6.74·13-s − 8.82·15-s − 4.92·17-s + 1.24·19-s − 2.18·21-s + 23-s + 11.3·25-s + 2.70·27-s + 6.48·29-s + 8.38·31-s − 8.69·33-s + 4.04·35-s − 7.61·37-s − 14.7·39-s − 7.11·41-s + 1.11·43-s + 7.12·45-s − 8.31·47-s + 49-s + 10.7·51-s − 4.77·53-s + 16.1·55-s + ⋯ |
| L(s) = 1 | − 1.25·3-s + 1.80·5-s + 0.377·7-s + 0.587·9-s + 1.20·11-s + 1.87·13-s − 2.27·15-s − 1.19·17-s + 0.286·19-s − 0.476·21-s + 0.208·23-s + 2.26·25-s + 0.519·27-s + 1.20·29-s + 1.50·31-s − 1.51·33-s + 0.683·35-s − 1.25·37-s − 2.35·39-s − 1.11·41-s + 0.170·43-s + 1.06·45-s − 1.21·47-s + 0.142·49-s + 1.50·51-s − 0.655·53-s + 2.17·55-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
| \(L(1)\) |
\(\approx\) |
\(2.079602238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.079602238\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.18T + 3T^{2} \) |
| 5 | \( 1 - 4.04T + 5T^{2} \) |
| 11 | \( 1 - 3.98T + 11T^{2} \) |
| 13 | \( 1 - 6.74T + 13T^{2} \) |
| 17 | \( 1 + 4.92T + 17T^{2} \) |
| 19 | \( 1 - 1.24T + 19T^{2} \) |
| 29 | \( 1 - 6.48T + 29T^{2} \) |
| 31 | \( 1 - 8.38T + 31T^{2} \) |
| 37 | \( 1 + 7.61T + 37T^{2} \) |
| 41 | \( 1 + 7.11T + 41T^{2} \) |
| 43 | \( 1 - 1.11T + 43T^{2} \) |
| 47 | \( 1 + 8.31T + 47T^{2} \) |
| 53 | \( 1 + 4.77T + 53T^{2} \) |
| 59 | \( 1 + 1.31T + 59T^{2} \) |
| 61 | \( 1 - 0.838T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 6.46T + 73T^{2} \) |
| 79 | \( 1 - 6.45T + 79T^{2} \) |
| 83 | \( 1 - 5.59T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−8.828495426036660830521125811639, −8.499979255508265293608474099680, −6.68236924624628484365837898750, −6.50515995795762076442701863139, −5.96838992219158596929642264730, −5.12230437209510040427105791994, −4.43116123299048469709298383456, −3.09428044609623225338367025990, −1.71062813981433036553603361517, −1.10804516502079385272903582764,
1.10804516502079385272903582764, 1.71062813981433036553603361517, 3.09428044609623225338367025990, 4.43116123299048469709298383456, 5.12230437209510040427105791994, 5.96838992219158596929642264730, 6.50515995795762076442701863139, 6.68236924624628484365837898750, 8.499979255508265293608474099680, 8.828495426036660830521125811639
