| L(s) = 1 | − 4i·7-s + 3·9-s − 4·11-s − 2i·13-s + 2i·17-s + 4·19-s − 4i·23-s − 2·29-s − 8·31-s − 6i·37-s − 6·41-s − 8i·43-s + 4i·47-s − 9·49-s + 6i·53-s + ⋯ |
| L(s) = 1 | − 1.51i·7-s + 9-s − 1.20·11-s − 0.554i·13-s + 0.485i·17-s + 0.917·19-s − 0.834i·23-s − 0.371·29-s − 1.43·31-s − 0.986i·37-s − 0.937·41-s − 1.21i·43-s + 0.583i·47-s − 1.28·49-s + 0.824i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.276868154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.276868154\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−9.270231862591860240656425884260, −8.140003332005757446198756522555, −7.39807223353356363562229469537, −7.10564226554235055524056984019, −5.83135606837318926877023171140, −4.92161818042559725828230006055, −4.06390688273939225570304188525, −3.24119016684682002831995332564, −1.79457675974558589641265003896, −0.48710251976651989983071855223,
1.63815028387202988186110531264, 2.61279107285671708396311829716, 3.62665358801877072391497326764, 5.03320658751984636300447285701, 5.33151221877542772152168056433, 6.43396732214031063076156644361, 7.36430741880210061266120750282, 8.017586346526220315568453851134, 9.017917827723684456430935223940, 9.572093029991495062249867714706
