| L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s + 308.·5-s − 216·6-s + 343·7-s − 512·8-s + 729·9-s − 2.46e3·10-s + 4.38e3·11-s + 1.72e3·12-s − 2.19e3·13-s − 2.74e3·14-s + 8.33e3·15-s + 4.09e3·16-s − 3.17e4·17-s − 5.83e3·18-s + 1.86e3·19-s + 1.97e4·20-s + 9.26e3·21-s − 3.50e4·22-s − 1.82e4·23-s − 1.38e4·24-s + 1.71e4·25-s + 1.75e4·26-s + 1.96e4·27-s + 2.19e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.10·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.781·10-s + 0.992·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.637·15-s + 0.250·16-s − 1.56·17-s − 0.235·18-s + 0.0622·19-s + 0.552·20-s + 0.218·21-s − 0.702·22-s − 0.313·23-s − 0.204·24-s + 0.220·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.952266181\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.952266181\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 3 | \( 1 - 27T \) |
| 7 | \( 1 - 343T \) |
| 13 | \( 1 + 2.19e3T \) |
| good | 5 | \( 1 - 308.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 4.38e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 3.17e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.86e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.82e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.36e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.21e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.40e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 9.34e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.58e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.67e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.31e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.12e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.91e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.68e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.09e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 7.43e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 7.91e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 9.45e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.24e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.63e7T + 8.07e13T^{2} \) |
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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
−9.629009653996160353826492886262, −8.814605717731772282270246377684, −8.207333922562955783533776168967, −6.87719175385207116347705511482, −6.39194404332491320261038790386, −5.09230089211930299325869000360, −3.92579251772812122737352586741, −2.48110024372493266460048391098, −1.88783668324006555734472652742, −0.816065115529438827972349334857,
0.816065115529438827972349334857, 1.88783668324006555734472652742, 2.48110024372493266460048391098, 3.92579251772812122737352586741, 5.09230089211930299325869000360, 6.39194404332491320261038790386, 6.87719175385207116347705511482, 8.207333922562955783533776168967, 8.814605717731772282270246377684, 9.629009653996160353826492886262
