| L(s) = 1 | + 1.41·2-s + 2.00·4-s + 3.16i·5-s + 2.82·8-s + 4.47i·10-s + 13.2·11-s − 5.49i·13-s + 4.00·16-s + 13.5i·17-s + 0.717i·19-s + 6.33i·20-s + 18.7·22-s + 2.27·23-s + 14.9·25-s − 7.76i·26-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s + 0.633i·5-s + 0.353·8-s + 0.447i·10-s + 1.20·11-s − 0.422i·13-s + 0.250·16-s + 0.797i·17-s + 0.0377i·19-s + 0.316i·20-s + 0.851·22-s + 0.0987·23-s + 0.598·25-s − 0.298i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.205123749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.205123749\) |
| \(L(2)\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3.16iT - 25T^{2} \) |
| 11 | \( 1 - 13.2T + 121T^{2} \) |
| 13 | \( 1 + 5.49iT - 169T^{2} \) |
| 17 | \( 1 - 13.5iT - 289T^{2} \) |
| 19 | \( 1 - 0.717iT - 361T^{2} \) |
| 23 | \( 1 - 2.27T + 529T^{2} \) |
| 29 | \( 1 + 20.4T + 841T^{2} \) |
| 31 | \( 1 - 24.6iT - 961T^{2} \) |
| 37 | \( 1 - 64.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 21.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 6.48T + 1.84e3T^{2} \) |
| 47 | \( 1 - 47.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 22.0T + 2.80e3T^{2} \) |
| 59 | \( 1 - 83.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 66.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 92.6T + 4.48e3T^{2} \) |
| 71 | \( 1 - 48.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 130. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 76.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 107. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 167. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 25.5iT - 9.40e3T^{2} \) |
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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
−10.17792666999374121761573742701, −9.271261484360067556326074924102, −8.247031828665671357174749329006, −7.26156321726418988666413864839, −6.46901925759053958925735196214, −5.79619945109085466932967662210, −4.56309047817362692800040330197, −3.67986816245922128510924033765, −2.73686412209988003438707155128, −1.34265330843990702305445438404,
0.952834671346080340450906598792, 2.27697421837978023070347830913, 3.64265849885300303281753492053, 4.45323018494937787248372736260, 5.32585581576011778826923744071, 6.33723419292844737019515519918, 7.08857239898925283578000204292, 8.114660816416562755350025284234, 9.170509657324535087910491458445, 9.633450226446577778224790283591
