| L(s) = 1 | − 1.73i·3-s − 2.23·5-s + 6.33i·7-s − 2.99·9-s + 9.27i·11-s − 18.5·13-s + 3.87i·15-s + 13.9·17-s − 17.2i·19-s + 10.9·21-s − 33.7i·23-s + 5.00·25-s + 5.19i·27-s + 28.6·29-s − 23.4i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.447·5-s + 0.904i·7-s − 0.333·9-s + 0.843i·11-s − 1.42·13-s + 0.258i·15-s + 0.818·17-s − 0.907i·19-s + 0.522·21-s − 1.46i·23-s + 0.200·25-s + 0.192i·27-s + 0.986·29-s − 0.757i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.156587303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.156587303\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 + 2.23T \) |
| good | 7 | \( 1 - 6.33iT - 49T^{2} \) |
| 11 | \( 1 - 9.27iT - 121T^{2} \) |
| 13 | \( 1 + 18.5T + 169T^{2} \) |
| 17 | \( 1 - 13.9T + 289T^{2} \) |
| 19 | \( 1 + 17.2iT - 361T^{2} \) |
| 23 | \( 1 + 33.7iT - 529T^{2} \) |
| 29 | \( 1 - 28.6T + 841T^{2} \) |
| 31 | \( 1 + 23.4iT - 961T^{2} \) |
| 37 | \( 1 - 67.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 44.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + 50.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 31.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 81.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 19.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 53.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.49iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 13.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 40.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 141. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 69.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 46.3T + 7.92e3T^{2} \) |
| 97 | \( 1 - 68.5T + 9.40e3T^{2} \) |
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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
−9.601016536826550029350721106667, −8.710941412509096778698001919485, −7.87382846025675447513979715217, −7.14661046538944716289049954904, −6.32737918692719051204854512503, −5.17473421844326536781202489335, −4.46482740500929414896513339405, −2.86427786003574437718585228359, −2.17049585924333193810913307847, −0.42564272124677550874536050716,
1.08343738117363926729833908671, 2.93073745730350223265265847765, 3.73936792643024632912806634484, 4.69737068196486354600512637441, 5.57080409331448718463140374731, 6.69440308926108277611281536690, 7.74436042142662450842942149886, 8.134791399350189221235232941368, 9.508505859985420971223772493606, 9.935455294606589068571377532048
