| L(s) = 1 | + 6.12i·5-s + (−6.21 + 3.23i)7-s + 15.2·11-s − 3.00i·13-s − 21.6i·17-s − 11.8i·19-s + 1.72·23-s − 12.5·25-s + 41.1·29-s + 8.50i·31-s + (−19.7 − 38.0i)35-s + 53.1·37-s − 43.0i·41-s − 36.6·43-s − 30.2i·47-s + ⋯ |
| L(s) = 1 | + 1.22i·5-s + (−0.887 + 0.461i)7-s + 1.38·11-s − 0.231i·13-s − 1.27i·17-s − 0.626i·19-s + 0.0748·23-s − 0.502·25-s + 1.41·29-s + 0.274i·31-s + (−0.565 − 1.08i)35-s + 1.43·37-s − 1.04i·41-s − 0.852·43-s − 0.643i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.998328150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.998328150\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (6.21 - 3.23i)T \) |
| good | 5 | \( 1 - 6.12iT - 25T^{2} \) |
| 11 | \( 1 - 15.2T + 121T^{2} \) |
| 13 | \( 1 + 3.00iT - 169T^{2} \) |
| 17 | \( 1 + 21.6iT - 289T^{2} \) |
| 19 | \( 1 + 11.8iT - 361T^{2} \) |
| 23 | \( 1 - 1.72T + 529T^{2} \) |
| 29 | \( 1 - 41.1T + 841T^{2} \) |
| 31 | \( 1 - 8.50iT - 961T^{2} \) |
| 37 | \( 1 - 53.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 43.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 36.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + 30.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 30T + 2.80e3T^{2} \) |
| 59 | \( 1 + 73.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 5.67iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 28.7T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.53T + 5.04e3T^{2} \) |
| 73 | \( 1 - 94.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 81.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 86.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 27.6iT - 7.92e3T^{2} \) |
| 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
−9.180275607828274706133566349134, −8.329660081806255086030552353671, −7.03985467458893543399313103382, −6.82561511201013303462258173170, −6.10540080750660311248302031688, −5.04649888114696999149918132601, −3.88747831614494902386876252464, −3.04954941575607994365127427130, −2.41236766670092126340101105777, −0.74286232753534657756103365232,
0.821559711646996471546777694608, 1.59983847299408451594721310964, 3.16610350105803896935249857609, 4.15106673703767906168109012338, 4.57939185829023736605867149108, 5.99191688308656488041364378757, 6.31842586812085483882264994975, 7.34653591695423409650995043537, 8.375149720910431651750273326688, 8.826756019110733177418027144660
