| L(s) = 1 | + 1.41·2-s − 1.73·3-s + 2.00·4-s − 6.00i·5-s − 2.44·6-s − 4.69i·7-s + 2.82·8-s + 2.99·9-s − 8.49i·10-s − 6.16i·11-s − 3.46·12-s + 8.71·13-s − 6.63i·14-s + 10.4i·15-s + 4.00·16-s − 14.0i·17-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.500·4-s − 1.20i·5-s − 0.408·6-s − 0.670i·7-s + 0.353·8-s + 0.333·9-s − 0.849i·10-s − 0.560i·11-s − 0.288·12-s + 0.670·13-s − 0.473i·14-s + 0.693i·15-s + 0.250·16-s − 0.828i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 + 0.902i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.50101 - 0.945995i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50101 - 0.945995i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 + 1.73T \) |
| 23 | \( 1 + (-9.92 - 20.7i)T \) |
| good | 5 | \( 1 + 6.00iT - 25T^{2} \) |
| 7 | \( 1 + 4.69iT - 49T^{2} \) |
| 11 | \( 1 + 6.16iT - 121T^{2} \) |
| 13 | \( 1 - 8.71T + 169T^{2} \) |
| 17 | \( 1 + 14.0iT - 289T^{2} \) |
| 19 | \( 1 - 14.6iT - 361T^{2} \) |
| 29 | \( 1 - 1.08T + 841T^{2} \) |
| 31 | \( 1 + 48.1T + 961T^{2} \) |
| 37 | \( 1 - 56.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 22.8T + 1.68e3T^{2} \) |
| 43 | \( 1 - 47.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 7.39T + 2.20e3T^{2} \) |
| 53 | \( 1 + 86.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 87.6T + 3.48e3T^{2} \) |
| 61 | \( 1 - 0.0196iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 104. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 7.78T + 5.04e3T^{2} \) |
| 73 | \( 1 - 19.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 46.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 16.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 88.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 38.8iT - 9.40e3T^{2} \) |
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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
−12.95695047862722603275116471631, −11.80664537561476314073603454353, −11.06046644325758051706920555257, −9.780643980688612903719092246624, −8.485086875374507436292135048869, −7.20440273058037939462808479024, −5.83544498857028496469689960625, −4.91390588843757083508342185081, −3.68489176956110196846085532309, −1.14587955974451556654637142139,
2.37340243130380816544956110358, 3.90615092375147097577436057985, 5.46019648930675252124303975478, 6.45878721859011674467518915749, 7.35033247188494305682844288640, 9.024375734004038270779010982581, 10.61242365486668416207905852652, 10.99602382143839315690950638896, 12.20841985756601906029436124546, 12.97838749698725006933469950134
