| L(s) = 1 | − 6.19i·2-s + 89.9i·3-s + 217.·4-s + 140. i·5-s + 557.·6-s − 4.21e3i·7-s − 2.93e3i·8-s − 1.52e3·9-s + 872.·10-s − 4.22e3·11-s + 1.95e4i·12-s + 3.53e4·13-s − 2.61e4·14-s − 1.26e4·15-s + 3.75e4·16-s + 2.30e4·17-s + ⋯ |
| L(s) = 1 | − 0.387i·2-s + 1.11i·3-s + 0.849·4-s + 0.225i·5-s + 0.430·6-s − 1.75i·7-s − 0.716i·8-s − 0.232·9-s + 0.0872·10-s − 0.288·11-s + 0.943i·12-s + 1.23·13-s − 0.680·14-s − 0.250·15-s + 0.572·16-s + 0.276·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.44413 - 0.366732i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.44413 - 0.366732i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-3.26e6 - 1.00e6i)T \) |
| good | 2 | \( 1 + 6.19iT - 256T^{2} \) |
| 3 | \( 1 - 89.9iT - 6.56e3T^{2} \) |
| 5 | \( 1 - 140. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 4.21e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 4.22e3T + 2.14e8T^{2} \) |
| 13 | \( 1 - 3.53e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 2.30e4T + 6.97e9T^{2} \) |
| 19 | \( 1 - 1.75e3iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 1.05e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 8.07e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 3.74e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 2.09e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 - 3.35e6T + 7.98e12T^{2} \) |
| 47 | \( 1 + 3.56e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + 5.72e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 1.72e7T + 1.46e14T^{2} \) |
| 61 | \( 1 + 1.48e7iT - 1.91e14T^{2} \) |
| 67 | \( 1 - 1.67e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 1.93e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 1.18e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 2.09e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 1.44e6T + 2.25e15T^{2} \) |
| 89 | \( 1 + 5.12e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 7.77e7T + 7.83e15T^{2} \) |
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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
−14.22363502855677002935209912027, −12.99798754333866272721004383126, −11.21105283507710784597562120532, −10.61049417335860080235789005573, −9.803268462166043450822092114203, −7.76691600017287948219710921023, −6.43143149079570617133748469938, −4.32877846357322469051817660525, −3.30655716109600654747146163280, −1.09346899731175003901002504101,
1.46616431157382584197890701007, 2.68167760676288915695701240967, 5.59346384714061362630672613490, 6.47708459838371557937232429390, 7.85150738047728545441811895122, 8.890622495538315713105700091473, 10.98167143810102653725699971733, 12.16497700636294861068998409057, 12.79290166130959796764197958695, 14.35483339615164462062082996511
