| L(s) = 1 | − 3.59i·3-s − 7.02i·5-s − 0.845i·7-s − 3.91·9-s − 14.5·11-s − 15.5·13-s − 25.2·15-s + 6.95·17-s + 30.8i·19-s − 3.03·21-s − 17.5·23-s − 24.3·25-s − 18.2i·27-s − 7.86i·29-s + 57.7·31-s + ⋯ |
| L(s) = 1 | − 1.19i·3-s − 1.40i·5-s − 0.120i·7-s − 0.435·9-s − 1.32·11-s − 1.19·13-s − 1.68·15-s + 0.409·17-s + 1.62i·19-s − 0.144·21-s − 0.761·23-s − 0.972·25-s − 0.676i·27-s − 0.271i·29-s + 1.86·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 - 0.855i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6368900458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6368900458\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + (22.2 + 36.7i)T \) |
| good | 3 | \( 1 + 3.59iT - 9T^{2} \) |
| 5 | \( 1 + 7.02iT - 25T^{2} \) |
| 7 | \( 1 + 0.845iT - 49T^{2} \) |
| 11 | \( 1 + 14.5T + 121T^{2} \) |
| 13 | \( 1 + 15.5T + 169T^{2} \) |
| 17 | \( 1 - 6.95T + 289T^{2} \) |
| 19 | \( 1 - 30.8iT - 361T^{2} \) |
| 23 | \( 1 + 17.5T + 529T^{2} \) |
| 29 | \( 1 + 7.86iT - 841T^{2} \) |
| 31 | \( 1 - 57.7T + 961T^{2} \) |
| 37 | \( 1 + 32.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 18.4T + 1.68e3T^{2} \) |
| 47 | \( 1 - 25.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 79.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 18.4T + 3.48e3T^{2} \) |
| 61 | \( 1 - 76.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 9.00T + 4.48e3T^{2} \) |
| 71 | \( 1 + 51.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 77.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 7.04T + 6.24e3T^{2} \) |
| 83 | \( 1 + 83.8T + 6.88e3T^{2} \) |
| 89 | \( 1 + 91.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 155.T + 9.40e3T^{2} \) |
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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
−9.807330685420039937497885963625, −8.479792241107911920133474029401, −7.932456600457618855567066361849, −7.33452123060819468197716279384, −6.02120697402676172615250221029, −5.24243311854378481084110983566, −4.24919108002996658355951848515, −2.52145593732797323357151255778, −1.44145718990025503762320195007, −0.22145742284189302759044630366,
2.59143522347690738423842833428, 3.10140772756418676482117799098, 4.53283236873946172771914214501, 5.17499555225673900639297339639, 6.46604109743880412339005483498, 7.32402911358353690946608515953, 8.209505981434878451808685549961, 9.547030712022799192136158022512, 10.05089005925476301592099771172, 10.64030179910674866973067223562
