L(s) = 1 | + 0.896i·2-s + 3.19·4-s − 8.66·5-s − 3.20i·7-s + 6.45i·8-s − 7.76i·10-s − 17.7i·13-s + 2.87·14-s + 7·16-s − 12.9i·17-s + 34.4i·19-s − 27.6·20-s + 10.7·23-s + 50.0·25-s + 15.9·26-s + ⋯ |
L(s) = 1 | + 0.448i·2-s + 0.799·4-s − 1.73·5-s − 0.458i·7-s + 0.806i·8-s − 0.776i·10-s − 1.36i·13-s + 0.205·14-s + 0.437·16-s − 0.762i·17-s + 1.81i·19-s − 1.38·20-s + 0.466·23-s + 2.00·25-s + 0.612·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.072831786\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.072831786\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 0.896iT - 4T^{2} \) |
| 5 | \( 1 + 8.66T + 25T^{2} \) |
| 7 | \( 1 + 3.20iT - 49T^{2} \) |
| 13 | \( 1 + 17.7iT - 169T^{2} \) |
| 17 | \( 1 + 12.9iT - 289T^{2} \) |
| 19 | \( 1 - 34.4iT - 361T^{2} \) |
| 23 | \( 1 - 10.7T + 529T^{2} \) |
| 29 | \( 1 - 34.5iT - 841T^{2} \) |
| 31 | \( 1 + 8.53T + 961T^{2} \) |
| 37 | \( 1 + 35.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 40.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 22.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 9.89T + 2.20e3T^{2} \) |
| 53 | \( 1 + 23.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 96.4T + 3.48e3T^{2} \) |
| 61 | \( 1 - 61.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 68.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 12.8T + 5.04e3T^{2} \) |
| 73 | \( 1 - 20.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 109. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 30.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 125.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 46.8T + 9.40e3T^{2} \) |
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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
−10.25623251620544084099841128152, −8.784969770311211623210647748822, −7.897474832119428165042946500237, −7.62240007942100699032699297896, −6.87161057858807661666244940796, −5.75866180061970508796326920092, −4.81172679530103090772952424543, −3.60506681733607766916962640806, −2.99763105381253383567706259814, −1.16147741928614257476793925862,
0.35524460394352115492525244248, 1.93794518816095063464213074576, 3.05919462113053597853912407766, 3.95768902470994993263876543102, 4.75190700734027067292923062734, 6.22402955668533862765081128017, 7.05721459099929860554711529549, 7.56523073201770988684389009854, 8.665112211630768626582685458601, 9.246329400310317683287876194666