| L(s) = 1 | + 1.73i·3-s + 1.01i·5-s + (−2.24 + 6.63i)7-s − 2.99·9-s + 10.2·11-s − 8.95i·13-s − 1.75·15-s − 30.4i·17-s − 16.1i·19-s + (−11.4 − 3.88i)21-s − 6.72·23-s + 23.9·25-s − 5.19i·27-s − 30·29-s − 50.1i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.202i·5-s + (−0.320 + 0.947i)7-s − 0.333·9-s + 0.931·11-s − 0.689i·13-s − 0.117·15-s − 1.78i·17-s − 0.849i·19-s + (−0.546 − 0.184i)21-s − 0.292·23-s + 0.958·25-s − 0.192i·27-s − 1.03·29-s − 1.61i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.687641542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.687641542\) |
| \(L(2)\) |
|
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 | \( 1 + (2.24 - 6.63i)T \) |
| good | 5 | \( 1 - 1.01iT - 25T^{2} \) |
| 11 | \( 1 - 10.2T + 121T^{2} \) |
| 13 | \( 1 + 8.95iT - 169T^{2} \) |
| 17 | \( 1 + 30.4iT - 289T^{2} \) |
| 19 | \( 1 + 16.1iT - 361T^{2} \) |
| 23 | \( 1 + 6.72T + 529T^{2} \) |
| 29 | \( 1 + 30T + 841T^{2} \) |
| 31 | \( 1 + 50.1iT - 961T^{2} \) |
| 37 | \( 1 + 30.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 7.10iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 74.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 58.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 70.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 0.492iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 2.86iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 27.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 50.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 70.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 133.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 104. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 144. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 100. iT - 9.40e3T^{2} \) |
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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
−9.184713090472598375297905312227, −9.022192977439176019805275621917, −7.74214331935143757285834702029, −6.90538071532030950529003269902, −5.93864914040933580507524077031, −5.22761425890529568408641677636, −4.25483166334737959394378101191, −3.13027046736749548229225698171, −2.39881828328938271194995289728, −0.55984898959412874445499493216,
1.08140311246764014210826153927, 1.91466173292914956807787410677, 3.61061627619485367508097786250, 4.04754869534046844597741964626, 5.38663160419658452083264319643, 6.42976525497889567505344173096, 6.87047098964398344051488362152, 7.80980742574037714630490007103, 8.657270154389072539664007517621, 9.322307660349564846794108815049
