| L(s) = 1 | − 2.28·2-s + 3.23·4-s + 4.23·5-s − 2.82·8-s − 9.69·10-s + 3.70·11-s + 2.28·13-s + 5·17-s − 5.45·19-s + 13.7·20-s − 8.47·22-s − 0.333·23-s + 12.9·25-s − 5.23·26-s + 7.19·29-s + 3.36·31-s + 5.65·32-s − 11.4·34-s − 4.70·37-s + 12.4·38-s − 11.9·40-s − 4.70·41-s + 2.23·43-s + 11.9·44-s + 0.763·46-s − 1.47·47-s − 29.6·50-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 1.61·4-s + 1.89·5-s − 0.999·8-s − 3.06·10-s + 1.11·11-s + 0.634·13-s + 1.21·17-s − 1.25·19-s + 3.06·20-s − 1.80·22-s − 0.0696·23-s + 2.58·25-s − 1.02·26-s + 1.33·29-s + 0.605·31-s + 1.00·32-s − 1.96·34-s − 0.774·37-s + 2.02·38-s − 1.89·40-s − 0.735·41-s + 0.340·43-s + 1.80·44-s + 0.112·46-s − 0.214·47-s − 4.18·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.252038408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.252038408\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.28T + 2T^{2} \) |
| 5 | \( 1 - 4.23T + 5T^{2} \) |
| 11 | \( 1 - 3.70T + 11T^{2} \) |
| 13 | \( 1 - 2.28T + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 + 5.45T + 19T^{2} \) |
| 23 | \( 1 + 0.333T + 23T^{2} \) |
| 29 | \( 1 - 7.19T + 29T^{2} \) |
| 31 | \( 1 - 3.36T + 31T^{2} \) |
| 37 | \( 1 + 4.70T + 37T^{2} \) |
| 41 | \( 1 + 4.70T + 41T^{2} \) |
| 43 | \( 1 - 2.23T + 43T^{2} \) |
| 47 | \( 1 + 1.47T + 47T^{2} \) |
| 53 | \( 1 + 3.16T + 53T^{2} \) |
| 59 | \( 1 + 3.94T + 59T^{2} \) |
| 61 | \( 1 + 3.70T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 + 0.746T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 0.236T + 83T^{2} \) |
| 89 | \( 1 + 1.70T + 89T^{2} \) |
| 97 | \( 1 + 1.95T + 97T^{2} \) |
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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
−9.642090809499828819277427538003, −8.815752423433399635122875170357, −8.508953665632759134884345411451, −7.22720138982762387780213196967, −6.34505159661455297220667457763, −6.01599573298195131709137351758, −4.64697112936517927580279808110, −2.99710637502269660851331122298, −1.81373188029910758595425340552, −1.18674478056210034752297285636,
1.18674478056210034752297285636, 1.81373188029910758595425340552, 2.99710637502269660851331122298, 4.64697112936517927580279808110, 6.01599573298195131709137351758, 6.34505159661455297220667457763, 7.22720138982762387780213196967, 8.508953665632759134884345411451, 8.815752423433399635122875170357, 9.642090809499828819277427538003
