| L(s) = 1 | + 1.41i·2-s + 1.73·3-s − 2.00·4-s + (−2.40 − 4.38i)5-s + 2.44i·6-s + (6.94 + 0.853i)7-s − 2.82i·8-s + 2.99·9-s + (6.20 − 3.39i)10-s + 2.88·11-s − 3.46·12-s + 13.8·13-s + (−1.20 + 9.82i)14-s + (−4.16 − 7.59i)15-s + 4.00·16-s + 24.1·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.500·4-s + (−0.480 − 0.876i)5-s + 0.408i·6-s + (0.992 + 0.121i)7-s − 0.353i·8-s + 0.333·9-s + (0.620 − 0.339i)10-s + 0.261·11-s − 0.288·12-s + 1.06·13-s + (−0.0861 + 0.701i)14-s + (−0.277 − 0.506i)15-s + 0.250·16-s + 1.42·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.85088 + 0.355232i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.85088 + 0.355232i\) |
| \(L(2)\) |
|
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 | \( 1 + (2.40 + 4.38i)T \) |
| 7 | \( 1 + (-6.94 - 0.853i)T \) |
| good | 11 | \( 1 - 2.88T + 121T^{2} \) |
| 13 | \( 1 - 13.8T + 169T^{2} \) |
| 17 | \( 1 - 24.1T + 289T^{2} \) |
| 19 | \( 1 - 6.53iT - 361T^{2} \) |
| 23 | \( 1 + 28.8iT - 529T^{2} \) |
| 29 | \( 1 + 32.9T + 841T^{2} \) |
| 31 | \( 1 + 2.43iT - 961T^{2} \) |
| 37 | \( 1 - 50.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 21.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 13.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 40.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 17.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 1.47iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 111. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 120. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 90.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 21.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 66.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + 78.5T + 6.88e3T^{2} \) |
| 89 | \( 1 + 90.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 44.1T + 9.40e3T^{2} \) |
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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
−12.34303708081590026713843676501, −11.36502716051275161306389561187, −10.00986478550606740050421503060, −8.762183061227765554301287702544, −8.291725709816683416461926721430, −7.43019643528688901985566834040, −5.87967276999592686030381966973, −4.75773299461799150109245955772, −3.66611310268841853931348902813, −1.32905926768523403191957811108,
1.55422702158172108871095695631, 3.18952039242103800300050060939, 4.07378958513816258385143068568, 5.67181018883846306129991879550, 7.34695362621771460226068223061, 8.069025657159938593366236911896, 9.193503583536141227229224662560, 10.30923953380164930463041105344, 11.21218338557724315644790526175, 11.77139230355915187498150070607
