| L(s) = 1 | − 1.41·2-s + 1.73i·3-s + 2.00·4-s − 2.44i·6-s + (−6.70 − 2.02i)7-s − 2.82·8-s − 2.99·9-s + 11.9·11-s + 3.46i·12-s + 9.67i·13-s + (9.47 + 2.86i)14-s + 4.00·16-s − 7.09i·17-s + 4.24·18-s − 17.5i·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577i·3-s + 0.500·4-s − 0.408i·6-s + (−0.957 − 0.288i)7-s − 0.353·8-s − 0.333·9-s + 1.08·11-s + 0.288i·12-s + 0.744i·13-s + (0.676 + 0.204i)14-s + 0.250·16-s − 0.417i·17-s + 0.235·18-s − 0.923i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.288 - 0.957i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.062888794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.062888794\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (6.70 + 2.02i)T \) |
| good | 11 | \( 1 - 11.9T + 121T^{2} \) |
| 13 | \( 1 - 9.67iT - 169T^{2} \) |
| 17 | \( 1 + 7.09iT - 289T^{2} \) |
| 19 | \( 1 + 17.5iT - 361T^{2} \) |
| 23 | \( 1 - 2.84T + 529T^{2} \) |
| 29 | \( 1 + 13.6T + 841T^{2} \) |
| 31 | \( 1 - 19.9iT - 961T^{2} \) |
| 37 | \( 1 - 11.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 28.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 72.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 28.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 11.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 101. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 76.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 76.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 95.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 120. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 14.8T + 6.24e3T^{2} \) |
| 83 | \( 1 - 60.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 88.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 18.8iT - 9.40e3T^{2} \) |
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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
−9.575985832220646214497803998014, −9.317697172032297048952838959029, −8.549293389415126057500977980936, −7.23256988383615407815933191648, −6.72327285134761105068327510234, −5.80038515941729671660395384347, −4.48798565055696076125840555710, −3.60904765728848879930151098936, −2.51728829150346201716101604502, −0.932413424656025726836538266237,
0.54096486868563034207791953386, 1.80564640982243801770293031143, 3.01342140541293651404846415531, 3.99116823602758365247667723961, 5.74293808248834953148885371631, 6.18984639633160760317824985391, 7.13691332101963245223735005932, 7.88626149300181794081958658364, 8.810489105010766116662372460814, 9.444277695900812665558779836416
