| L(s) = 1 | − 14i·11-s − 2·17-s + 34i·19-s − 25·25-s − 46·41-s + 14i·43-s + 49·49-s + 82i·59-s − 62i·67-s + 142·73-s + 158i·83-s + 146·89-s − 94·97-s + 178i·107-s − 98·113-s + ⋯ |
| L(s) = 1 | − 1.27i·11-s − 0.117·17-s + 1.78i·19-s − 25-s − 1.12·41-s + 0.325i·43-s + 0.999·49-s + 1.38i·59-s − 0.925i·67-s + 1.94·73-s + 1.90i·83-s + 1.64·89-s − 0.969·97-s + 1.66i·107-s − 0.867·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.210914795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.210914795\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 25T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 + 14iT - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 + 2T + 289T^{2} \) |
| 19 | \( 1 - 34iT - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 + 46T + 1.68e3T^{2} \) |
| 43 | \( 1 - 14iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 - 82iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 3.72e3T^{2} \) |
| 67 | \( 1 + 62iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 142T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 158iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 146T + 7.92e3T^{2} \) |
| 97 | \( 1 + 94T + 9.40e3T^{2} \) |
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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
−8.939284526295777653591433512383, −8.190530568249052015763585354561, −7.68553469683379495791268773040, −6.54465743575403063118973098033, −5.90221571558033534348119472734, −5.22125493170035461934639503032, −3.96733518129112844468538075317, −3.41468901446744630226797826368, −2.20685126292063679483799303089, −1.05064677977929055866508205068,
0.31900895163668774927211558194, 1.79017803258175259125737971965, 2.63264535510755924963645641131, 3.80854533031571788080379725504, 4.68386481388771039323818546795, 5.31862575349472687776961389778, 6.47487971182028025804306296428, 7.06960589325945655193877622589, 7.77289402342246750620084690312, 8.717955870305394497696191875309
