| L(s) = 1 | + 1.41i·2-s − 1.73·3-s − 2.00·4-s − 6.71·5-s − 2.44i·6-s − 6.32·7-s − 2.82i·8-s + 2.99·9-s − 9.50i·10-s + 8.40i·11-s + 3.46·12-s − 7.72i·13-s − 8.94i·14-s + 11.6·15-s + 4.00·16-s + 18.2·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.500·4-s − 1.34·5-s − 0.408i·6-s − 0.903·7-s − 0.353i·8-s + 0.333·9-s − 0.950i·10-s + 0.764i·11-s + 0.288·12-s − 0.593i·13-s − 0.638i·14-s + 0.775·15-s + 0.250·16-s + 1.07·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.102i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.688482 + 0.0353597i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.688482 + 0.0353597i\) |
| \(L(2)\) |
|
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 | \( 1 + 1.73T \) |
| 59 | \( 1 + (-6.04 - 58.6i)T \) |
| good | 5 | \( 1 + 6.71T + 25T^{2} \) |
| 7 | \( 1 + 6.32T + 49T^{2} \) |
| 11 | \( 1 - 8.40iT - 121T^{2} \) |
| 13 | \( 1 + 7.72iT - 169T^{2} \) |
| 17 | \( 1 - 18.2T + 289T^{2} \) |
| 19 | \( 1 - 10.8T + 361T^{2} \) |
| 23 | \( 1 + 3.86iT - 529T^{2} \) |
| 29 | \( 1 - 23.6T + 841T^{2} \) |
| 31 | \( 1 + 30.7iT - 961T^{2} \) |
| 37 | \( 1 + 6.95iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 15.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 1.63iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 2.26iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 4.04T + 2.80e3T^{2} \) |
| 61 | \( 1 + 81.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 33.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 106.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 30.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 22.6T + 6.24e3T^{2} \) |
| 83 | \( 1 - 117. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 91.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 14.5iT - 9.40e3T^{2} \) |
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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
−11.39713136904429629451323804096, −10.22460664599852562133745645541, −9.480601791622889971918204705179, −8.090531338392385250137685097423, −7.47961779879524824611624318396, −6.55852928661987192844269768899, −5.45451013320178858773323139819, −4.32492404741850147727421918128, −3.27986351130738764733631482801, −0.51426500961607994011535735808,
0.863654922163368877094173431982, 3.12619789378853902835689577538, 3.89126427021358915941464325004, 5.14524467684636761957421229122, 6.40568042621712104301099508378, 7.50549590493082124044993641815, 8.502001019657394032757447518098, 9.589551769138782477613591922453, 10.49595965746986332835476143501, 11.42133964613211034825989418073
