L(s) = 1 | − 27·3-s − 127.·5-s − 547.·7-s + 729·9-s − 5.22e3·11-s − 1.10e4·13-s + 3.43e3·15-s + 3.56e3·17-s − 3.78e4·19-s + 1.47e4·21-s + 3.18e3·23-s − 6.19e4·25-s − 1.96e4·27-s + 3.65e3·29-s − 1.81e5·31-s + 1.41e5·33-s + 6.96e4·35-s + 1.33e5·37-s + 2.97e5·39-s − 1.89e5·41-s − 4.33e5·43-s − 9.28e4·45-s − 1.27e5·47-s − 5.24e5·49-s − 9.61e4·51-s − 1.84e6·53-s + 6.65e5·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.455·5-s − 0.603·7-s + 0.333·9-s − 1.18·11-s − 1.39·13-s + 0.262·15-s + 0.175·17-s − 1.26·19-s + 0.348·21-s + 0.0546·23-s − 0.792·25-s − 0.192·27-s + 0.0278·29-s − 1.09·31-s + 0.683·33-s + 0.274·35-s + 0.434·37-s + 0.802·39-s − 0.430·41-s − 0.831·43-s − 0.151·45-s − 0.179·47-s − 0.636·49-s − 0.101·51-s − 1.70·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.03357397390\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03357397390\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 27T \) |
good | 5 | \( 1 + 127.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 547.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.22e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.10e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.56e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.78e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.18e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 3.65e3T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.81e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.33e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.89e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.33e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.27e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.84e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 4.49e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.57e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.13e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.53e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.45e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.20e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.08e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.79e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.99e6T + 8.07e13T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.21855737068131802407715993657, −9.443952608861145170298424155060, −8.112294367345583308441449975182, −7.36870430490234901285583273472, −6.36528240750520531786090191091, −5.30204884396201608139563361442, −4.42407594599404673328467001412, −3.13067269673774481599959378820, −1.97171917795230638677856483641, −0.081814977589238039949477988126,
0.081814977589238039949477988126, 1.97171917795230638677856483641, 3.13067269673774481599959378820, 4.42407594599404673328467001412, 5.30204884396201608139563361442, 6.36528240750520531786090191091, 7.36870430490234901285583273472, 8.112294367345583308441449975182, 9.443952608861145170298424155060, 10.21855737068131802407715993657