| L(s) = 1 | − 58.8i·5-s − 3.23e3·7-s + 3.24e3i·11-s − 4.22e3·13-s − 1.30e5i·17-s + 6.22e4·19-s + 1.71e5i·23-s + 3.87e5·25-s + 1.39e5i·29-s + 6.23e5·31-s + 1.90e5i·35-s + 7.69e5·37-s + 3.32e6i·41-s + 2.13e6·43-s − 4.35e6i·47-s + ⋯ |
| L(s) = 1 | − 0.0941i·5-s − 1.34·7-s + 0.221i·11-s − 0.147·13-s − 1.56i·17-s + 0.477·19-s + 0.611i·23-s + 0.991·25-s + 0.197i·29-s + 0.675·31-s + 0.126i·35-s + 0.410·37-s + 1.17i·41-s + 0.625·43-s − 0.892i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.2035713866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2035713866\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 58.8iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 3.23e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 3.24e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 4.22e3T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.30e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 6.22e4T + 1.69e10T^{2} \) |
| 23 | \( 1 - 1.71e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 1.39e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 6.23e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 7.69e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 3.32e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 2.13e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 4.35e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 1.45e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 6.44e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 8.38e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 5.73e6T + 4.06e14T^{2} \) |
| 71 | \( 1 - 3.38e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 8.92e6T + 8.06e14T^{2} \) |
| 79 | \( 1 + 6.06e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 2.96e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 3.08e6iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 9.27e7T + 7.83e15T^{2} \) |
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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
−9.611827977912765716454504727445, −8.579774552113829598689483307608, −7.30889795603740966518472435097, −6.71822166816537890943145398309, −5.60274306712566566969885376372, −4.61069369536863095086138125561, −3.31535236240372597754651479475, −2.62285653878440217689987992991, −1.04119215783362022083522687709, −0.04545218778209985561021886070,
1.10634139317481317694673065570, 2.55660011916713082777419374975, 3.42569963306791283633954247856, 4.44746267066893376123032254343, 5.86549308587539662993248534105, 6.43615745040044413461478287729, 7.46344704967870256825668465492, 8.563657285327006484936056821226, 9.385377964285142013554918436311, 10.29408175409768972256391526885
