| L(s) = 1 | − 1.73i·3-s + 0.417·5-s − 2.99·9-s + 18.4i·11-s + 1.16·13-s − 0.723i·15-s − 0.417·17-s − 21.1i·19-s + 16.9i·23-s − 24.8·25-s + 5.19i·27-s + 4.33·29-s − 20.7i·31-s + 31.9·33-s − 61.1·37-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.0834·5-s − 0.333·9-s + 1.67i·11-s + 0.0896·13-s − 0.0482i·15-s − 0.0245·17-s − 1.11i·19-s + 0.738i·23-s − 0.993·25-s + 0.192i·27-s + 0.149·29-s − 0.670i·31-s + 0.967·33-s − 1.65·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6939095426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6939095426\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 0.417T + 25T^{2} \) |
| 11 | \( 1 - 18.4iT - 121T^{2} \) |
| 13 | \( 1 - 1.16T + 169T^{2} \) |
| 17 | \( 1 + 0.417T + 289T^{2} \) |
| 19 | \( 1 + 21.1iT - 361T^{2} \) |
| 23 | \( 1 - 16.9iT - 529T^{2} \) |
| 29 | \( 1 - 4.33T + 841T^{2} \) |
| 31 | \( 1 + 20.7iT - 961T^{2} \) |
| 37 | \( 1 + 61.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 9.07T + 1.68e3T^{2} \) |
| 43 | \( 1 - 7.30iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 19.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 92.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 99.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 78.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 77.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 43.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 53.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 74.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 32.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 81.9T + 7.92e3T^{2} \) |
| 97 | \( 1 - 30.1T + 9.40e3T^{2} \) |
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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
−8.427251776165363770904698036016, −7.49824616549153332317428287264, −7.10781182033392105934296085316, −6.27441119755722482957328493416, −5.29700892015897448170576304935, −4.56134398525228240916183845795, −3.52987317870867121183152822677, −2.31751462900379023057927650987, −1.65205393825054313573273720860, −0.16982063156988279293594728730,
1.17436538884681563788121685552, 2.57207772985668609739763713094, 3.52548102150223635396273207390, 4.13559220699278787562756502109, 5.37841282198820942285490473684, 5.82282639174542888148321607909, 6.68037338475128260962351005599, 7.75461871322066235253043288860, 8.578366384180062155618568931136, 8.895752616844363872678636431190
