| L(s) = 1 | + 12.7·2-s − 27·3-s + 34.5·4-s − 425.·5-s − 344.·6-s + 1.13e3·7-s − 1.19e3·8-s + 729·9-s − 5.42e3·10-s − 932.·12-s − 6.59e3·13-s + 1.44e4·14-s + 1.14e4·15-s − 1.96e4·16-s − 1.57e4·17-s + 9.29e3·18-s − 2.12e4·19-s − 1.46e4·20-s − 3.06e4·21-s − 4.35e4·23-s + 3.21e4·24-s + 1.02e5·25-s − 8.40e4·26-s − 1.96e4·27-s + 3.92e4·28-s + 8.87e4·29-s + 1.46e5·30-s + ⋯ |
| L(s) = 1 | + 1.12·2-s − 0.577·3-s + 0.269·4-s − 1.52·5-s − 0.650·6-s + 1.25·7-s − 0.822·8-s + 0.333·9-s − 1.71·10-s − 0.155·12-s − 0.832·13-s + 1.41·14-s + 0.878·15-s − 1.19·16-s − 0.777·17-s + 0.375·18-s − 0.712·19-s − 0.410·20-s − 0.722·21-s − 0.747·23-s + 0.474·24-s + 1.31·25-s − 0.937·26-s − 0.192·27-s + 0.337·28-s + 0.676·29-s + 0.989·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.215308454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.215308454\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 12.7T + 128T^{2} \) |
| 5 | \( 1 + 425.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.13e3T + 8.23e5T^{2} \) |
| 13 | \( 1 + 6.59e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.57e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.12e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.35e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 8.87e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 3.13e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.58e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.47e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.05e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.54e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.48e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.98e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.43e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.17e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.61e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.69e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.40e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.75e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.64e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 9.08e6T + 8.07e13T^{2} \) |
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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
−10.83850549611257321569532376205, −9.201083176258129204203362788004, −8.128699477420845855102239196939, −7.37724497374555944958778928055, −6.19328338905384616392559643929, −4.95035866339314027451579591401, −4.49180937191508630942410224974, −3.68351727285483705519483468737, −2.18227747602148719040875368459, −0.43317232819016698139423845186,
0.43317232819016698139423845186, 2.18227747602148719040875368459, 3.68351727285483705519483468737, 4.49180937191508630942410224974, 4.95035866339314027451579591401, 6.19328338905384616392559643929, 7.37724497374555944958778928055, 8.128699477420845855102239196939, 9.201083176258129204203362788004, 10.83850549611257321569532376205
