| L(s) = 1 | − 2.23·5-s − 0.763i·7-s + 5.70i·23-s + 5.00·25-s − 6·29-s + 1.70i·35-s + 4.47·41-s + 11.2i·43-s + 13.7i·47-s + 6.41·49-s − 13.4·61-s + 8.18i·67-s + 17.7i·83-s + 6·89-s + 18·101-s + ⋯ |
| L(s) = 1 | − 0.999·5-s − 0.288i·7-s + 1.19i·23-s + 1.00·25-s − 1.11·29-s + 0.288i·35-s + 0.698·41-s + 1.71i·43-s + 1.99i·47-s + 0.916·49-s − 1.71·61-s + 0.999i·67-s + 1.94i·83-s + 0.635·89-s + 1.79·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8844060155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8844060155\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| good | 7 | \( 1 + 0.763iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 5.70iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 - 11.2iT - 43T^{2} \) |
| 47 | \( 1 - 13.7iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 - 8.18iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 17.7iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.586164339750300985145667727421, −9.003148411219530185346330034822, −7.79346287836696513965330525748, −7.62886311933258916854510515325, −6.55526190724563738863186979149, −5.57816554126748423400448726305, −4.52923907614230466791568109167, −3.78889472190645049311628637546, −2.82538235659231648077356102088, −1.23629850438442708890282226677,
0.39156806227144678853467505921, 2.12072966994198831819881376486, 3.31906518039096453568612848383, 4.16114513807988639728471297203, 5.06810304912545001160237796950, 6.05787805785935546198358658898, 7.05451846298634568720811072803, 7.67272406988954743806120204033, 8.616405583855486674573862430536, 9.078370359371748802396301680386
