| L(s) = 1 | + 8·2-s − 4.29·3-s + 64·4-s − 487.·5-s − 34.3·6-s + 343·7-s + 512·8-s − 2.16e3·9-s − 3.89e3·10-s − 2.42e3·11-s − 274.·12-s − 2.19e3·13-s + 2.74e3·14-s + 2.09e3·15-s + 4.09e3·16-s − 1.38e3·17-s − 1.73e4·18-s + 3.97e4·19-s − 3.11e4·20-s − 1.47e3·21-s − 1.93e4·22-s + 22.9·23-s − 2.19e3·24-s + 1.59e5·25-s − 1.75e4·26-s + 1.87e4·27-s + 2.19e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0918·3-s + 0.5·4-s − 1.74·5-s − 0.0649·6-s + 0.377·7-s + 0.353·8-s − 0.991·9-s − 1.23·10-s − 0.548·11-s − 0.0459·12-s − 0.277·13-s + 0.267·14-s + 0.160·15-s + 0.250·16-s − 0.0681·17-s − 0.701·18-s + 1.32·19-s − 0.871·20-s − 0.0347·21-s − 0.388·22-s + 0.000392·23-s − 0.0324·24-s + 2.03·25-s − 0.196·26-s + 0.182·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.746078076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.746078076\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 7 | \( 1 - 343T \) |
| 13 | \( 1 + 2.19e3T \) |
| good | 3 | \( 1 + 4.29T + 2.18e3T^{2} \) |
| 5 | \( 1 + 487.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 2.42e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 1.38e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.97e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 22.9T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.52e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.81e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.33e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.22e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.00e6T + 2.71e11T^{2} \) |
| 47 | \( 1 + 6.27e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.95e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.25e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.15e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.14e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.27e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.60e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.21e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.72e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.00e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.33e6T + 8.07e13T^{2} \) |
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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
−11.37478969265848175737041631402, −10.98845978009991880441745299447, −9.223484241299854337253422184779, −7.80775153684007705676262882542, −7.54700419599683187850410567780, −5.85678937794736327644109837618, −4.78487306332019921162229015487, −3.72122976920686279667807392937, −2.68223761641362730445391457697, −0.63270519955703317367709165747,
0.63270519955703317367709165747, 2.68223761641362730445391457697, 3.72122976920686279667807392937, 4.78487306332019921162229015487, 5.85678937794736327644109837618, 7.54700419599683187850410567780, 7.80775153684007705676262882542, 9.223484241299854337253422184779, 10.98845978009991880441745299447, 11.37478969265848175737041631402
