| L(s) = 1 | + 2-s − 1.41i·3-s + 4-s + 1.41i·5-s − 1.41i·6-s + 8-s + 0.999·9-s + 1.41i·10-s − 2.82i·11-s − 1.41i·12-s + 2·13-s + 2.00·15-s + 16-s + (3 + 2.82i)17-s + 0.999·18-s − 4·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.816i·3-s + 0.5·4-s + 0.632i·5-s − 0.577i·6-s + 0.353·8-s + 0.333·9-s + 0.447i·10-s − 0.852i·11-s − 0.408i·12-s + 0.554·13-s + 0.516·15-s + 0.250·16-s + (0.727 + 0.685i)17-s + 0.235·18-s − 0.917·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.061178514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.061178514\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 17 | \( 1 + (-3 - 2.82i)T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 1.41iT - 3T^{2} \) |
| 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 7.07iT - 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 4.24iT - 31T^{2} \) |
| 37 | \( 1 - 8.48iT - 37T^{2} \) |
| 41 | \( 1 + 2.82iT - 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 61 | \( 1 + 8.48iT - 61T^{2} \) |
| 67 | \( 1 - 14T + 67T^{2} \) |
| 71 | \( 1 + 11.3iT - 71T^{2} \) |
| 73 | \( 1 + 4.24iT - 73T^{2} \) |
| 79 | \( 1 + 8.48iT - 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 4.24iT - 97T^{2} \) |
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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
−8.739368808341882661276862037341, −8.209495776356524227356731749493, −7.29224330289164538197250249342, −6.42414568976893805704297897705, −6.27264441684113499328945716849, −5.06168464564981376894882801995, −4.02279395271207158886241399573, −3.17357419649774508008846264252, −2.20487729694860312858244866789, −1.03225919074893835135744675664,
1.27497619216047795474960636252, 2.53256842118922009663334651894, 3.86382045219950488255191986257, 4.20293017756040656648274052972, 5.19167933875192561546021203654, 5.68658250302459622809080923319, 6.92697351519564258103361389147, 7.52795155958275918945649534278, 8.570973083145581782998812872989, 9.433574081830349060915977396450
