| L(s) = 1 | + 1.16i·3-s − 4.97i·7-s + 1.63·9-s − 11-s − 0.665i·13-s − 6.77i·17-s − 19-s + 5.80·21-s + 2.16i·23-s + 5.41i·27-s − 7.97·29-s − 8.94·31-s − 1.16i·33-s + 0.139i·37-s + 0.776·39-s + ⋯ |
| L(s) = 1 | + 0.674i·3-s − 1.87i·7-s + 0.545·9-s − 0.301·11-s − 0.184i·13-s − 1.64i·17-s − 0.229·19-s + 1.26·21-s + 0.451i·23-s + 1.04i·27-s − 1.48·29-s − 1.60·31-s − 0.203i·33-s + 0.0229i·37-s + 0.124·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.056003629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.056003629\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 1.16iT - 3T^{2} \) |
| 7 | \( 1 + 4.97iT - 7T^{2} \) |
| 13 | \( 1 + 0.665iT - 13T^{2} \) |
| 17 | \( 1 + 6.77iT - 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 2.16iT - 23T^{2} \) |
| 29 | \( 1 + 7.97T + 29T^{2} \) |
| 31 | \( 1 + 8.94T + 31T^{2} \) |
| 37 | \( 1 - 0.139iT - 37T^{2} \) |
| 41 | \( 1 + 1.80T + 41T^{2} \) |
| 43 | \( 1 - 2.80iT - 43T^{2} \) |
| 47 | \( 1 + 0.530iT - 47T^{2} \) |
| 53 | \( 1 + 6.30iT - 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 9.60iT - 67T^{2} \) |
| 71 | \( 1 + 9.80T + 71T^{2} \) |
| 73 | \( 1 + 7.02iT - 73T^{2} \) |
| 79 | \( 1 + 5.50T + 79T^{2} \) |
| 83 | \( 1 - 13.5iT - 83T^{2} \) |
| 89 | \( 1 - 9.58T + 89T^{2} \) |
| 97 | \( 1 - 14.7iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.077865311042405410828134779355, −7.68415748643580166580605690998, −7.41345721259849185292513483744, −6.69187255166458058538889037253, −5.36601259370412771891157760842, −4.70140895929062324763487119829, −3.88833330362496669967535139511, −3.27587536915220414749518283516, −1.67404407520385136011375059001, −0.34902508411128516001583502466,
1.72746763159072244422342267147, 2.20055758678322310219710062903, 3.44778923961065800279526637817, 4.52563964430970735548335112865, 5.72250857918311024026354904560, 5.95515826705176332163779117372, 7.00025536660538152592965640395, 7.78468147275289457068348027696, 8.610830981180129953149968106631, 9.053904100395233743747480488148
