| L(s) = 1 | − 1.41i·3-s + 2.23i·5-s + (1.58 − 2.12i)7-s + 0.999·9-s + 3.16·15-s + (−3 − 2.23i)21-s + 9.48·23-s − 5.00·25-s − 5.65i·27-s + 6·29-s + (4.74 + 3.53i)35-s − 4.47i·41-s − 3.16·43-s + 2.23i·45-s − 9.89i·47-s + ⋯ |
| L(s) = 1 | − 0.816i·3-s + 0.999i·5-s + (0.597 − 0.801i)7-s + 0.333·9-s + 0.816·15-s + (−0.654 − 0.487i)21-s + 1.97·23-s − 1.00·25-s − 1.08i·27-s + 1.11·29-s + (0.801 + 0.597i)35-s − 0.698i·41-s − 0.482·43-s + 0.333i·45-s − 1.44i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.801 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.55563 - 0.515972i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.55563 - 0.515972i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 + (-1.58 + 2.12i)T \) |
| good | 3 | \( 1 + 1.41iT - 3T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 9.48T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 4.47iT - 41T^{2} \) |
| 43 | \( 1 + 3.16T + 43T^{2} \) |
| 47 | \( 1 + 9.89iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 13.4iT - 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 15.5iT - 83T^{2} \) |
| 89 | \( 1 - 17.8iT - 89T^{2} \) |
| 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
−10.65940323241133269210255991682, −10.08430261231104218476912854264, −8.763322279057286576209557189770, −7.69604200085216329688910369519, −7.08305536119470653045395387305, −6.50354176585888331883918880074, −5.08022283917042703253124464199, −3.87774861442806613281747660106, −2.58650491845040053861014156606, −1.19519006487133455947345797481,
1.43369861357658652537926170817, 3.09815116358423415539328696436, 4.63387379531427056522861992042, 4.87796898194416701105327363594, 6.05453309090106003980353915514, 7.41642984694329464742011246067, 8.500428163462022435066111299549, 9.080210946927347610033782606986, 9.830657342098631065512589578494, 10.85590556732596628815457743481
