| L(s) = 1 | + 1.73i·3-s + 2·5-s − 2.99·9-s + 17.5i·11-s − 20.3·13-s + 3.46i·15-s − 32.3·17-s + 28.0i·19-s − 10.2i·23-s − 21·25-s − 5.19i·27-s + 22.6·29-s − 27.7i·31-s − 30.3·33-s + 62.6·37-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.400·5-s − 0.333·9-s + 1.59i·11-s − 1.56·13-s + 0.230i·15-s − 1.90·17-s + 1.47i·19-s − 0.443i·23-s − 0.839·25-s − 0.192i·27-s + 0.781·29-s − 0.893i·31-s − 0.919·33-s + 1.69·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3603130026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3603130026\) |
| \(L(2)\) |
|
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 | \( 1 - 2T + 25T^{2} \) |
| 11 | \( 1 - 17.5iT - 121T^{2} \) |
| 13 | \( 1 + 20.3T + 169T^{2} \) |
| 17 | \( 1 + 32.3T + 289T^{2} \) |
| 19 | \( 1 - 28.0iT - 361T^{2} \) |
| 23 | \( 1 + 10.2iT - 529T^{2} \) |
| 29 | \( 1 - 22.6T + 841T^{2} \) |
| 31 | \( 1 + 27.7iT - 961T^{2} \) |
| 37 | \( 1 - 62.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 28.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + 48.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 91.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 46.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 41.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 12.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 55.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 53.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 21.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 96.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 63.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 16.9T + 7.92e3T^{2} \) |
| 97 | \( 1 - 95.3T + 9.40e3T^{2} \) |
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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
−8.758558271328595667527583654604, −7.78756782696303280346001794193, −7.10096099814647227539035628645, −6.27362513205963156016408311196, −5.33334501240991896594506632917, −4.48428076603122448311248108415, −4.05595992396231138366721597447, −2.40907478784877800961567709071, −2.07904537905960221966469301763, −0.092324230491916657908148268011,
0.982614205373243722268732483066, 2.44972344785513905765459147608, 2.81849004310528102582249356419, 4.31796573525379661887647197241, 5.05662332463734613098709469025, 6.08259443163057832027408241112, 6.57734651597606124680347777449, 7.45835877345775560706489657469, 8.182464492999096467714670509108, 9.116187905834588827251434880694
