L(s) = 1 | − 0.192i·2-s + 3.96·4-s + 7.62i·5-s − 2.45·7-s − 1.53i·8-s + 1.46·10-s + 3.31i·11-s + 20.4·13-s + 0.473i·14-s + 15.5·16-s − 14.2i·17-s − 25.2·19-s + 30.2i·20-s + 0.638·22-s − 7.64i·23-s + ⋯ |
L(s) = 1 | − 0.0963i·2-s + 0.990·4-s + 1.52i·5-s − 0.350·7-s − 0.191i·8-s + 0.146·10-s + 0.301i·11-s + 1.57·13-s + 0.0338i·14-s + 0.972·16-s − 0.841i·17-s − 1.32·19-s + 1.51i·20-s + 0.0290·22-s − 0.332i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.50460 + 0.478220i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.50460 + 0.478220i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 - 3.31iT \) |
good | 2 | \( 1 + 0.192iT - 4T^{2} \) |
| 5 | \( 1 - 7.62iT - 25T^{2} \) |
| 7 | \( 1 + 2.45T + 49T^{2} \) |
| 13 | \( 1 - 20.4T + 169T^{2} \) |
| 17 | \( 1 + 14.2iT - 289T^{2} \) |
| 19 | \( 1 + 25.2T + 361T^{2} \) |
| 23 | \( 1 + 7.64iT - 529T^{2} \) |
| 29 | \( 1 + 43.4iT - 841T^{2} \) |
| 31 | \( 1 + 33.4T + 961T^{2} \) |
| 37 | \( 1 - 24.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 21.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 17.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 76.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 3.42iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 68.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 31.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 30.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 102. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 131.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 10.6T + 6.24e3T^{2} \) |
| 83 | \( 1 - 126. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 114. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 109.T + 9.40e3T^{2} \) |
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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
−13.87223171907757204735243815500, −12.63354722760201055357540406395, −11.19243726455338647234754776822, −10.92339468688349341162247648328, −9.719713642601990320223996199029, −7.961001581959298424916835011078, −6.73112678231266129649095057373, −6.16492224891397776236856513674, −3.64566846317129982146168578188, −2.37018323971967355982322231806,
1.50825432679160940571810485086, 3.76791304914603876587605377418, 5.51666761476179908773503919558, 6.54989642386214829361046822516, 8.212133449231634621973819424220, 8.899587468459361129912280467307, 10.49242801914829279759395810245, 11.44975474641831683133811760732, 12.69616020941054263230023674858, 13.15951125683423259193436294684