| L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s − 479.·5-s − 216·6-s + 343·7-s − 512·8-s + 729·9-s + 3.83e3·10-s + 6.14e3·11-s + 1.72e3·12-s − 2.19e3·13-s − 2.74e3·14-s − 1.29e4·15-s + 4.09e3·16-s − 2.82e4·17-s − 5.83e3·18-s − 4.29e4·19-s − 3.06e4·20-s + 9.26e3·21-s − 4.91e4·22-s + 9.15e4·23-s − 1.38e4·24-s + 1.51e5·25-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 − 1.71·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 1.21·10-s + 1.39·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.990·15-s + 0.250·16-s − 1.39·17-s − 0.235·18-s − 1.43·19-s − 0.857·20-s + 0.218·21-s − 0.983·22-s + 1.56·23-s − 0.204·24-s + 1.94·25-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
| \(L(4)\) |
\(\approx\) |
\(1.117609511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.117609511\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 479.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 6.14e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 2.82e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.29e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 9.15e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 3.82e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.53e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.80e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 8.29e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.54e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 8.00e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.98e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.20e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.21e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.98e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.12e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 8.83e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.21e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.90e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.19e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 9.55e6T + 8.07e13T^{2} \) |
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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
−9.238478159810471028607058974884, −8.763290965649997375641106797834, −8.079416584175567709410654634823, −7.10328936536830803491538123499, −6.59314027193793905071250043930, −4.62468293954688111452919835125, −4.01744408505230889760830806018, −2.93517689400129079748890594403, −1.66075344592837576216082039097, −0.50018597728293339324545780929,
0.50018597728293339324545780929, 1.66075344592837576216082039097, 2.93517689400129079748890594403, 4.01744408505230889760830806018, 4.62468293954688111452919835125, 6.59314027193793905071250043930, 7.10328936536830803491538123499, 8.079416584175567709410654634823, 8.763290965649997375641106797834, 9.238478159810471028607058974884
