| L(s) = 1 | + 1.41i·2-s + (2.99 + 0.209i)3-s − 2.00·4-s − 5.00i·5-s + (−0.295 + 4.23i)6-s + 0.691·7-s − 2.82i·8-s + (8.91 + 1.25i)9-s + 7.07·10-s − 11.3i·11-s + (−5.98 − 0.418i)12-s − 0.232·13-s + 0.978i·14-s + (1.04 − 14.9i)15-s + 4.00·16-s − 4.25i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (0.997 + 0.0696i)3-s − 0.500·4-s − 1.00i·5-s + (−0.0492 + 0.705i)6-s + 0.0988·7-s − 0.353i·8-s + (0.990 + 0.139i)9-s + 0.707·10-s − 1.03i·11-s + (−0.498 − 0.0348i)12-s − 0.0178·13-s + 0.0698i·14-s + (0.0697 − 0.997i)15-s + 0.250·16-s − 0.250i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0696i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.28275 - 0.0796235i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.28275 - 0.0796235i\) |
| \(L(2)\) |
|
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 | \( 1 + (-2.99 - 0.209i)T \) |
| 59 | \( 1 - 7.68iT \) |
| good | 5 | \( 1 + 5.00iT - 25T^{2} \) |
| 7 | \( 1 - 0.691T + 49T^{2} \) |
| 11 | \( 1 + 11.3iT - 121T^{2} \) |
| 13 | \( 1 + 0.232T + 169T^{2} \) |
| 17 | \( 1 + 4.25iT - 289T^{2} \) |
| 19 | \( 1 - 21.0T + 361T^{2} \) |
| 23 | \( 1 + 5.79iT - 529T^{2} \) |
| 29 | \( 1 + 34.7iT - 841T^{2} \) |
| 31 | \( 1 - 4.00T + 961T^{2} \) |
| 37 | \( 1 - 28.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 55.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 48.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 33.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 38.5iT - 2.80e3T^{2} \) |
| 61 | \( 1 - 12.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 15.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 47.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 115.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 78.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 87.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 78.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 168.T + 9.40e3T^{2} \) |
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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
−11.27340622558548601587728013376, −9.833745870149647532917002366298, −9.225349394011119833015057713349, −8.285329695084315514978657877317, −7.82896691006249113670870164546, −6.49294102502931239453913141830, −5.25816100296736246388898022530, −4.29851630623418714677080339899, −3.00998356816273412211270600431, −1.06759549555382187899268060727,
1.69831375567193470417294931220, 2.85536864511248185989986648860, 3.74680018660719130238921880868, 5.05024452447997850584784919769, 6.79370832200706286412351728822, 7.50030883069298888325361894319, 8.593403243584591866782377247749, 9.649239837564851381863063744472, 10.19539396862181450778438287889, 11.17499225863491279939861583183
