| L(s) = 1 | + 8·2-s + 91.2·3-s + 64·4-s + 141.·5-s + 730.·6-s + 343·7-s + 512·8-s + 6.14e3·9-s + 1.12e3·10-s + 8.25e3·11-s + 5.84e3·12-s − 2.19e3·13-s + 2.74e3·14-s + 1.28e4·15-s + 4.09e3·16-s − 1.51e4·17-s + 4.91e4·18-s − 4.88e4·19-s + 9.03e3·20-s + 3.13e4·21-s + 6.60e4·22-s − 6.95e4·23-s + 4.67e4·24-s − 5.81e4·25-s − 1.75e4·26-s + 3.61e5·27-s + 2.19e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.95·3-s + 0.5·4-s + 0.505·5-s + 1.38·6-s + 0.377·7-s + 0.353·8-s + 2.81·9-s + 0.357·10-s + 1.87·11-s + 0.976·12-s − 0.277·13-s + 0.267·14-s + 0.985·15-s + 0.250·16-s − 0.747·17-s + 1.98·18-s − 1.63·19-s + 0.252·20-s + 0.737·21-s + 1.32·22-s − 1.19·23-s + 0.690·24-s − 0.744·25-s − 0.196·26-s + 3.53·27-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
| \(L(4)\) |
\(\approx\) |
\(8.239307782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.239307782\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 91.2T + 2.18e3T^{2} \) |
| 5 | \( 1 - 141.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 8.25e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 1.51e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.88e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.95e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.18e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.50e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.00e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.10e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.10e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 3.90e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.37e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.67e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.00e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.82e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.05e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.29e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.58e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.22e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.68e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.64e6T + 8.07e13T^{2} \) |
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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
−11.50965907334257064198666575219, −10.07051207881664588984736431128, −9.201233472199975339510471542499, −8.413246210025467837963938219689, −7.25078116023393760012660606621, −6.24514985717501031534652190936, −4.27351368644735083037385276936, −3.80290852840397713565171616099, −2.22615756762275061223509841324, −1.70931481254472736448362039874,
1.70931481254472736448362039874, 2.22615756762275061223509841324, 3.80290852840397713565171616099, 4.27351368644735083037385276936, 6.24514985717501031534652190936, 7.25078116023393760012660606621, 8.413246210025467837963938219689, 9.201233472199975339510471542499, 10.07051207881664588984736431128, 11.50965907334257064198666575219
