L(s) = 1 | + i·2-s − 4-s + (1.41 − 2.23i)7-s − i·8-s − 5.65i·11-s − 4.47i·13-s + (2.23 + 1.41i)14-s + 16-s + 3.16·17-s + 3.16i·19-s + 5.65·22-s + 4i·23-s + 4.47·26-s + (−1.41 + 2.23i)28-s + 2.82i·29-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (0.534 − 0.845i)7-s − 0.353i·8-s − 1.70i·11-s − 1.24i·13-s + (0.597 + 0.377i)14-s + 0.250·16-s + 0.766·17-s + 0.725i·19-s + 1.20·22-s + 0.834i·23-s + 0.877·26-s + (−0.267 + 0.422i)28-s + 0.525i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0515 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0515 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.290090439\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.290090439\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.41 + 2.23i)T \) |
good | 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 + 4.47iT - 13T^{2} \) |
| 17 | \( 1 - 3.16T + 17T^{2} \) |
| 19 | \( 1 - 3.16iT - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 6.32iT - 31T^{2} \) |
| 37 | \( 1 + 9.89T + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 - 1.41T + 43T^{2} \) |
| 47 | \( 1 + 9.48T + 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 + 9.48iT - 61T^{2} \) |
| 67 | \( 1 + 7.07T + 67T^{2} \) |
| 71 | \( 1 + 1.41iT - 71T^{2} \) |
| 73 | \( 1 + 13.4iT - 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 + 4.47T + 89T^{2} \) |
| 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
−8.186428712337322670847868500691, −7.902359316624373882776119084054, −7.13204165262505162639016893837, −6.09236479148614769576224694139, −5.57761224127837606265081884085, −4.86478677449348114851513785751, −3.57390549096476523198561226069, −3.31464349580203815806681797382, −1.46501368366665234254235831081, −0.40593017970357439573493476341,
1.56044560215701069714901800221, 2.13393710561386351087995336095, 3.11978808311078708443016738801, 4.36238325984634671526288526335, 4.77386809118354434494401045667, 5.60343251283410587479033250702, 6.79803518002509791303980261179, 7.25801540950600298837310554236, 8.494347331458185469785648614905, 8.775562264232799005458912803044