| L(s) = 1 | + (2 + i)5-s − 2i·7-s − 6·11-s + 2i·13-s + 6i·17-s + 4·19-s + 8i·23-s + (3 + 4i)25-s + 8·31-s + (2 − 4i)35-s + 2i·37-s + 6·41-s + 4i·43-s − 4i·47-s + 3·49-s + ⋯ |
| L(s) = 1 | + (0.894 + 0.447i)5-s − 0.755i·7-s − 1.80·11-s + 0.554i·13-s + 1.45i·17-s + 0.917·19-s + 1.66i·23-s + (0.600 + 0.800i)25-s + 1.43·31-s + (0.338 − 0.676i)35-s + 0.328i·37-s + 0.937·41-s + 0.609i·43-s − 0.583i·47-s + 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.659983449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.659983449\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2 - i)T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−9.918524403088762170473002579870, −8.963038262681087590633277980616, −7.82786276848491707873416013472, −7.41693938654424017512019829630, −6.30108283539833563984273186761, −5.62198478657392009764146007303, −4.72790412560141550769861125802, −3.52516413620517767042117659902, −2.56354587501872548344213573507, −1.39353661403997746584626956759,
0.68716604341194270866132660873, 2.53988248841054243364518118030, 2.72618260975546817422673726282, 4.64613829410457338332224649916, 5.28939251791783839832225971430, 5.83719539955991900639192361957, 6.94919713779200689862612012481, 7.934639018310698605278205367596, 8.581622453640456814619943474933, 9.448215772545647206191608867061
