| L(s) = 1 | − i·2-s − 3-s − 4-s − 3i·5-s + i·6-s + i·8-s + 9-s − 3·10-s + 5i·11-s + 12-s + (3 − 2i)13-s + 3i·15-s + 16-s − 3·17-s − i·18-s − i·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577·3-s − 0.5·4-s − 1.34i·5-s + 0.408i·6-s + 0.353i·8-s + 0.333·9-s − 0.948·10-s + 1.50i·11-s + 0.288·12-s + (0.832 − 0.554i)13-s + 0.774i·15-s + 0.250·16-s − 0.727·17-s − 0.235i·18-s − 0.229i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.347186945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.347186945\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (-3 + 2i)T \) |
| good | 5 | \( 1 + 3iT - 5T^{2} \) |
| 11 | \( 1 - 5iT - 11T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + iT - 19T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 7iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 14T + 53T^{2} \) |
| 59 | \( 1 - 14iT - 59T^{2} \) |
| 61 | \( 1 - 3T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 10iT - 71T^{2} \) |
| 73 | \( 1 - 11iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 16iT - 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
| 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
−8.657125988312815435372014826257, −7.77840636651054930316797343792, −6.88492420145800632857061378353, −5.96849787516930691220951283719, −5.12365964625832896332202733507, −4.55996690636076979084861311869, −4.02355196865025184086112996245, −2.66683924418493550889596527714, −1.59272909353369140486450796570, −0.813828937525400171009416244286,
0.62448721802453955512396882995, 2.19555752122016092262001852524, 3.38779819560221011376225504792, 3.91434705374179076335939784827, 5.06168442479569709608310555560, 5.95788415156388779443713366608, 6.37673355583598373350023667169, 6.89803646769211186237488301309, 7.72782540541938451114662998913, 8.582577335381216807710173931707
