| L(s) = 1 | + (−1 + i)2-s − 2i·4-s + (−2 − 2i)5-s + (2 + 2i)8-s + 3i·9-s + 4·10-s + (1 + i)11-s − 4·16-s + 2·17-s + (−3 − 3i)18-s + (−2 + 2i)19-s + (−4 + 4i)20-s − 2·22-s − 6i·23-s + 3i·25-s + ⋯ |
| L(s) = 1 | + (−0.707 + 0.707i)2-s − i·4-s + (−0.894 − 0.894i)5-s + (0.707 + 0.707i)8-s + i·9-s + 1.26·10-s + (0.301 + 0.301i)11-s − 16-s + 0.485·17-s + (−0.707 − 0.707i)18-s + (−0.458 + 0.458i)19-s + (−0.894 + 0.894i)20-s − 0.426·22-s − 1.25i·23-s + 0.600i·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.798004 - 0.158733i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.798004 - 0.158733i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1 - i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 3iT^{2} \) |
| 5 | \( 1 + (2 + 2i)T + 5iT^{2} \) |
| 11 | \( 1 + (-1 - i)T + 11iT^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + (2 - 2i)T - 19iT^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + (-7 + 7i)T - 29iT^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (5 + 5i)T + 37iT^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 + (1 + i)T + 43iT^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + (1 + i)T + 53iT^{2} \) |
| 59 | \( 1 + (8 + 8i)T + 59iT^{2} \) |
| 61 | \( 1 + (6 - 6i)T - 61iT^{2} \) |
| 67 | \( 1 + (-3 + 3i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + (-10 + 10i)T - 83iT^{2} \) |
| 89 | \( 1 - 14iT - 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−10.25659233632624403058140503210, −9.136509592257109452326992971534, −8.292922770104392256645978453947, −7.971842557862310803922103731323, −6.96382996137667195536448325336, −5.91864304319347871997126854932, −4.83270976941007915784579483180, −4.22729191304944437007389581291, −2.21925090854364058198112603049, −0.64909663662900655600071297757,
1.10861372974426628568882498339, 2.97820188139708549960425720950, 3.45090509881951208464321337002, 4.57010123587605586027017564728, 6.34127366071256277371655773187, 7.02703721455337747646231827117, 7.897046213596400062573097288406, 8.729460618470114343075716107426, 9.558580908218755588735998872242, 10.44111012520458151237336616860
