L(s) = 1 | + (0.5 + 0.866i)5-s + (0.5 + 0.866i)9-s + (−0.499 + 0.866i)25-s + 2·29-s − 2·41-s + (−0.499 + 0.866i)45-s + (1 + 1.73i)61-s + (−0.499 + 0.866i)81-s + (−1 − 1.73i)89-s + (−1 + 1.73i)101-s + (1 − 1.73i)109-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)5-s + (0.5 + 0.866i)9-s + (−0.499 + 0.866i)25-s + 2·29-s − 2·41-s + (−0.499 + 0.866i)45-s + (1 + 1.73i)61-s + (−0.499 + 0.866i)81-s + (−1 − 1.73i)89-s + (−1 + 1.73i)101-s + (1 − 1.73i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.458337947\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.458337947\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - 2T + T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + 2T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.658324089163570759377829810949, −8.076981934288995250842476762173, −7.12687349369153388765452661726, −6.75185943142755218652918484211, −5.84483935684594269645001444564, −5.06647583653431159683539426337, −4.27086280084149204760307857238, −3.19217669200412904210876527403, −2.43850170423643638790939319817, −1.48812474695699962345333133290,
0.883798214941200687861064139717, 1.85906446390295164884588257946, 3.03745872317234705303229426957, 3.99719055258620313241446925148, 4.78245619302691252748903274452, 5.43929766255881672517468222856, 6.45892390804888911059095724372, 6.78459089254537512914890994565, 7.976548706184446865390362053358, 8.544904331268213053730215793229