| L(s) = 1 | + 1.41i·2-s − 1.73·3-s − 2.00·4-s + (1.38 − 4.80i)5-s − 2.44i·6-s + (−5.24 + 4.63i)7-s − 2.82i·8-s + 2.99·9-s + (6.79 + 1.95i)10-s + 11.7·11-s + 3.46·12-s + 24.8·13-s + (−6.54 − 7.42i)14-s + (−2.39 + 8.32i)15-s + 4.00·16-s + 7.26·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.500·4-s + (0.276 − 0.961i)5-s − 0.408i·6-s + (−0.749 + 0.661i)7-s − 0.353i·8-s + 0.333·9-s + (0.679 + 0.195i)10-s + 1.06·11-s + 0.288·12-s + 1.90·13-s + (−0.467 − 0.530i)14-s + (−0.159 + 0.554i)15-s + 0.250·16-s + 0.427·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 - 0.428i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.903 - 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.27694 + 0.287647i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27694 + 0.287647i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 + (-1.38 + 4.80i)T \) |
| 7 | \( 1 + (5.24 - 4.63i)T \) |
| good | 11 | \( 1 - 11.7T + 121T^{2} \) |
| 13 | \( 1 - 24.8T + 169T^{2} \) |
| 17 | \( 1 - 7.26T + 289T^{2} \) |
| 19 | \( 1 + 23.0iT - 361T^{2} \) |
| 23 | \( 1 - 26.4iT - 529T^{2} \) |
| 29 | \( 1 - 57.0T + 841T^{2} \) |
| 31 | \( 1 - 10.2iT - 961T^{2} \) |
| 37 | \( 1 - 14.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 16.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 82.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 51.6T + 2.20e3T^{2} \) |
| 53 | \( 1 + 64.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 81.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 13.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 22.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 91.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 71.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 45.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 17.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 77.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 6.15T + 9.40e3T^{2} \) |
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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
−12.25761753326182247721749612239, −11.46975594112093068344507918403, −10.00665966901445367295581454653, −9.022970482141154955311230204166, −8.467465601154293612431591441393, −6.73332897078876340036845345478, −6.05745581389433347582273572479, −5.06606026485416099302057404773, −3.68846217975483150066415804849, −1.08655043194595114141011575119,
1.22024239900995042286166793687, 3.24740825352504877813851868148, 4.15125878165208119558655577803, 6.15265780506129878418592558574, 6.52832473752605864826373651937, 8.147591049601380366085558501787, 9.510511992925841157287314818620, 10.38578338008742413639010763504, 10.95707001032172888121461260865, 11.93631652866468471619619138441
