| L(s) = 1 | − i·3-s + i·7-s + 2·9-s + 3·11-s + 5i·13-s − 3i·17-s + 2·19-s + 21-s + 6i·23-s − 5i·27-s − 3·29-s + 4·31-s − 3i·33-s − 2i·37-s + 5·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.377i·7-s + 0.666·9-s + 0.904·11-s + 1.38i·13-s − 0.727i·17-s + 0.458·19-s + 0.218·21-s + 1.25i·23-s − 0.962i·27-s − 0.557·29-s + 0.718·31-s − 0.522i·33-s − 0.328i·37-s + 0.800·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.000520457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.000520457\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 5iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 - 9iT - 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 - iT - 97T^{2} \) |
| 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
−9.076483495131931179507310328039, −7.945496856904303250023343141820, −7.27605138985739995459645829336, −6.65890302773713973926079054717, −5.99737284483136288898452025132, −4.88204837624322250273005221277, −4.16738737846872918726561232221, −3.16049749778980608064662142910, −1.92964546349134719376424194392, −1.22415663766923704513892922976,
0.72769160946730103515556759342, 1.95807079822079965417446168077, 3.38008059864677102785436434291, 3.85675213672576188413657468274, 4.82269173143132563651173293933, 5.51866826379235964630934968779, 6.60138536640667112352505113218, 7.09831670611053340194324339248, 8.183899561835595921904247382750, 8.638473472508937174834678388002
