| L(s) = 1 | − 2i·3-s + (−1 − 2i)5-s + 2i·7-s − 9-s + 4·11-s + 4i·13-s + (−4 + 2i)15-s − 4·19-s + 4·21-s + 2i·23-s + (−3 + 4i)25-s − 4i·27-s − 2·29-s − 8i·33-s + (4 − 2i)35-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s + (−0.447 − 0.894i)5-s + 0.755i·7-s − 0.333·9-s + 1.20·11-s + 1.10i·13-s + (−1.03 + 0.516i)15-s − 0.917·19-s + 0.872·21-s + 0.417i·23-s + (−0.600 + 0.800i)25-s − 0.769i·27-s − 0.371·29-s − 1.39i·33-s + (0.676 − 0.338i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.789428 - 0.487893i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.789428 - 0.487893i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1 + 2i)T \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 4iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 14iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 8iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 16iT - 97T^{2} \) |
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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
−14.03027895351106023041294264280, −12.91997052424831310620284565891, −12.15060959609603369715962295845, −11.46522605854082072550973007185, −9.352390215483697261062375577735, −8.531576274232393632586704936129, −7.21583158999792306410434978903, −6.06383267017174124142765982170, −4.26729765342000317017177688222, −1.71634013614640719267961657709,
3.43992358269873926659198241096, 4.44387010686589718708577246840, 6.36127294369327897367988140029, 7.67898330238913181870458570066, 9.210189714894171573445856454892, 10.44721562698896132870423004877, 10.86241726330800794944868649596, 12.27661252309399260434141940896, 13.80403053219716984980398795584, 14.87516829368151963944878567818
