| L(s) = 1 | − 6.32·5-s + 2·7-s + 12.6·11-s + 16.7i·13-s + (9 + 16.7i)19-s − 6.32·23-s + 15.0·25-s − 21.1i·29-s + 16.7i·31-s − 12.6·35-s − 50.1i·37-s + 21.1i·41-s − 34·43-s + 82.2·47-s − 45·49-s + ⋯ |
| L(s) = 1 | − 1.26·5-s + 0.285·7-s + 1.14·11-s + 1.28i·13-s + (0.473 + 0.880i)19-s − 0.274·23-s + 0.600·25-s − 0.729i·29-s + 0.539i·31-s − 0.361·35-s − 1.35i·37-s + 0.516i·41-s − 0.790·43-s + 1.74·47-s − 0.918·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.098071809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.098071809\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-9 - 16.7i)T \) |
| good | 5 | \( 1 + 6.32T + 25T^{2} \) |
| 7 | \( 1 - 2T + 49T^{2} \) |
| 11 | \( 1 - 12.6T + 121T^{2} \) |
| 13 | \( 1 - 16.7iT - 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 23 | \( 1 + 6.32T + 529T^{2} \) |
| 29 | \( 1 + 21.1iT - 841T^{2} \) |
| 31 | \( 1 - 16.7iT - 961T^{2} \) |
| 37 | \( 1 + 50.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 21.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 34T + 1.84e3T^{2} \) |
| 47 | \( 1 - 82.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 63.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 21.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 86T + 3.72e3T^{2} \) |
| 67 | \( 1 - 66.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 126. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 102T + 5.32e3T^{2} \) |
| 79 | \( 1 - 117. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 25.2T + 6.88e3T^{2} \) |
| 89 | \( 1 - 126. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 33.4iT - 9.40e3T^{2} \) |
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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
−8.825327340347298663036223756777, −8.130854514924798633015789457060, −7.39244345439526766227245868706, −6.76022286349856580542363079912, −5.91166197553396410834968300021, −4.79166079622327055801876374749, −3.95310631873343617423378791985, −3.65403297737527322059683536036, −2.15235594742540907913175186086, −1.09025906377026233034097989215,
0.30540138204274482205725811347, 1.31213221796337293787367660065, 2.82933440500997712516385050159, 3.59875281124650053037433071774, 4.36175489445693716198396646950, 5.17732237737954275526170252390, 6.13869683484277312553454149825, 7.09224474658295613238650114209, 7.61058226210413232469325484660, 8.393943783486297289448173278975
