| L(s) = 1 | + 15.5i·3-s + 85.3·5-s + 511. i·7-s − 243·9-s − 1.21e3i·11-s + 2.74e3·13-s + 1.33e3i·15-s − 8.52e3·17-s − 1.04e4i·19-s − 7.97e3·21-s − 4.30e3i·23-s − 8.33e3·25-s − 3.78e3i·27-s + 4.24e4·29-s + 5.78e4i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.683·5-s + 1.49i·7-s − 0.333·9-s − 0.911i·11-s + 1.24·13-s + 0.394i·15-s − 1.73·17-s − 1.52i·19-s − 0.861·21-s − 0.353i·23-s − 0.533·25-s − 0.192i·27-s + 1.74·29-s + 1.94i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.563252736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.563252736\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 15.5iT \) |
| good | 5 | \( 1 - 85.3T + 1.56e4T^{2} \) |
| 7 | \( 1 - 511. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 1.21e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 2.74e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 8.52e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 1.04e4iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 4.30e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 4.24e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 5.78e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 8.35e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 7.01e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 2.87e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 5.87e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.11e5T + 2.21e10T^{2} \) |
| 59 | \( 1 - 1.18e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 5.31e4T + 5.15e10T^{2} \) |
| 67 | \( 1 - 4.21e4iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 3.51e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 9.12e4T + 1.51e11T^{2} \) |
| 79 | \( 1 - 7.75e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 6.69e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 5.04e5T + 4.96e11T^{2} \) |
| 97 | \( 1 - 1.03e6T + 8.32e11T^{2} \) |
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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
−9.347612163368251876545722730185, −8.775789112042352973852560660315, −8.444443904648009632196549146524, −6.60803298941009710678110825521, −6.11409289561630514512069587992, −5.23903861815814414764047840631, −4.31908993264425048157036234955, −2.91915003706714249078441763789, −2.35935143558040599630013088669, −0.901942502695670678060484760929,
0.56049155501122076392166278512, 1.48574060142063855871432474761, 2.35111295349325407412785175793, 3.88444473458662295089953985819, 4.46972937365938365262898494694, 6.03508542968240824249834607165, 6.43964418630607672971457763151, 7.50372066557115782708276437203, 8.106531054678168928383138019178, 9.288114914781608138995245431629
